Open Access

Differential inequalities imposed by the extended hypergeometric function

SpringerPlus20165:375

https://doi.org/10.1186/s40064-016-1996-9

Received: 15 September 2015

Accepted: 14 March 2016

Published: 25 March 2016

Abstract

Recently, the generalized hypergeometric function is extended by utilizing the Beta function. Based on this type of function, we introduce a new operator in the open unit disk. The present article investigates some subordination and superordination results for certain normalized analytic functions in the open unit disk, which are acted upon by the generalized Noor integral operator. Some of outcomes improve and generalize previously known outcomes.

Keywords

Analytic function Univalent function Integral operator Subordination Superordination Hypergeometric function Unit disk

Mathematics Subject Classification

30C45

Background

The theory of hypergeometric functions theaters significant and imposing role in the study of the fractional calculus and the geometric function theory. It motivates the theory of univalent functions, by attractive to current research after their utilization in the proof of great famous problem in geometric function theory which is called by the Bieberbach’s conjecture. This theory has been developed with enriched many presentations and simplification by protruding complex analysis.

Let \(H(\mathcal{U})\) be the class of all holomorphic functions \(\phi (z)\) which are defined in the unit disk \(\mathcal{U}\). For \(\alpha \in \mathbb {C}\) and \(n\in \mathbb {N}\), we let \(H \left[ a,n\right] =\{ {\phi \in H(\mathcal{U}):\phi (z)=a+a_{n} z^{n} +a_{n+1} z^{n+1}+\cdots }\}\) and \(\mathcal A\) be the subclass of \(H(\mathcal{U})\) consisting of functions of the form
$$\phi (z) = z + \sum _{n = 2}^\infty {a_n z^n},\quad (z\in \mathcal{U}).$$
(1)
For functions \(\phi (z)\), given (1), and \(\psi (z)\) given by
$$\psi (z) = z + \sum _{n = 2}^\infty {b_n z^n},\quad (z \in \mathcal{U}).$$
(2)
the Hadamard product (or convolution) of \(\phi (z)\) and \(\psi (z)\) is defined by
$$(\phi *\psi )(z) = z + \sum _{n = 2}^\infty {a_n b_n z^n},\quad (z \in \mathcal{U}).$$
(3)
For \(\phi\) and \(\varphi\) be members of the function class \(H(\mathcal{U})\), the function \(\phi\) is said to be subordinate to \(\varphi\), or \(\varphi\) is superordinate to \(\phi\), if there is an analytic function \(\theta (z)\) in \(\mathcal{U}\) with \(\theta (0)=0\) and \(|\theta (z)|<1\) for all \(z\in \mathcal{U}\), such that \(\phi (z)={\varphi {(\theta (z))}}\). In this case, we write \(\phi \prec \varphi ,\) or \(\phi (z)\prec \varphi (z).\) Furthermore, if the function \(\varphi\) is univalent in \(\mathcal{U},\) then we have the following equivalence (Miller and Mocanu 2000):
$$\phi (z)\prec \varphi (z)\quad (z\in \mathcal{U}) \iff \phi (0)=\varphi (0),\quad\phi (\mathcal{U})\subset \varphi (\mathcal{U}).$$
Let \(\phi :\mathbb {C}^2\longrightarrow \mathbb {C}\) and let \(\vartheta\) be univalent in \(z\in \mathcal{U}\). If \(\rho\) is analytic in \(\mathcal{U}\) and satisfies the differential subordination \(\phi (\rho (z),z\rho ^{\prime }(z))\prec \vartheta (z)\) then \(\rho\) is called a solution of the differential subordination. The univalent function \(\eta\) is called a dominant of the solutions of the differential subordination, \(\rho \prec \eta\). If \(\rho\) and \(\phi (\rho (z),z\rho ^{\prime }(z))\) are univalent in \(\mathcal{U}\) and satisfy the differential superordination \(\vartheta (z) \prec \phi (\rho (z),z\rho ^{\prime }(z))\) then \(\rho\) is called a solution of the differential superordination. An analytic \(\eta\) is called subordinate of the solution of the differential superordination if \(\eta \prec \rho\) (Miller and Mocanu 2003).
For real or complex numbers \(\alpha ,\beta ,\gamma\) other than \(0,-1,-2,\ldots\) the Gaussian hypergeometric function is defined by de Branges (1985)
$$F(\alpha ,\beta ;\gamma ,z)=\sum _{n=0}^{\infty }\frac{(\alpha )_n\,(\beta )_n}{(\gamma )_n\,(1)_n}\,z^n =1+\frac{\alpha \beta }{\gamma }z+\frac{\alpha (\alpha +1)\beta (\beta +1)}{\gamma (\gamma +1)}\frac{z^2}{2!}+\cdots$$
(4)
where \((\alpha )_n\) is the Pochhammer symbol defined by
$$(\alpha )_n=\frac{\Gamma (\alpha +n)}{\Gamma (\alpha )}=\left\{ \begin{array}{ll} 1,&\quad n=0\\ \alpha (\alpha +1)\ldots(\alpha +n-1), &\quad n=\{1,2,\ldots \} \end{array}\right.$$
and achieved
$$\begin{aligned}&(i)\,\,\, n(\wp )_n=\wp \left[ (\wp +1)_n-(\wp )_n\right] \\&(ii)\,\,\,(\wp )_{n+1}=\wp (\wp +1)_n=(\wp +n)(\wp )_n. \end{aligned}$$
Let B(xy) be the familiar Beta function defined by Srivastava et al. (2012, p. 8))
$$B(x,y)=\left\{ \begin{array}{ll} \int _{0}^{1}\sigma ^{x-1}(1-\sigma )^{y-1}d\sigma &\quad (\mathfrak {R}(x)>0;\mathfrak {R}(y)>0)\\ \frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)}&\quad (x,y \in \mathbb {C}\,\mathbb {Z}^-_0) \end{array}\right.$$
Here \(\Gamma\) denotes the Euler’s Gamma function (see, e.g., Srivastava and Choi 2012, Section 1.1). Srivastava et al. (2014) \(\mathcal{B}^{a,b;\kappa ,\mu }_p(x,y),\) introduced the extended Beta function as follows
$$\begin{aligned}&\mathcal{B}^{a,b;\kappa ,\mu }_p(x,y)=\int _{0}^{1}\sigma ^{x-1}(1-\sigma )^{y-1} F\left( a;b;-\frac{p}{\sigma ^{\kappa }(1-\sigma )^{\mu }}\right) d\sigma , \\&\quad\left( \kappa \ge 0, \mu \ge 0,\,\mathfrak {R}(p)\ge 0,\,\min \{\mathfrak {R}(a),\mathfrak {R}(b)\}>0, \mathfrak {R}(x)>-\mathfrak {R}(\kappa a), \mathfrak {R}(y)>-\mathfrak {R}(\mu a)\right) . \end{aligned}$$
(5)
The special case of (5) when \(p=0\) is seen to immediately reduce to the familiar beta function B(xy) (min \(\{\mathfrak {R}(x), \mathfrak {R}(y)\}>0\)) (Srivastava and Choi 2012).
Agarwal et al. (2015) introduced the extended Gauss hypergeometric function as follows
$$\begin{aligned}&F_{p;\kappa ,\mu }(\alpha ,\beta ;\gamma ;z;m):= \sum _{n=0}^{\infty }\frac{(\alpha )_n(\beta )_n}{(\gamma )_n}\frac{B^{a,b;\kappa , \mu }_p(\beta +n,\gamma -\beta +m)}{B(\beta +n,\gamma -\beta +m)}\frac{z^n}{n!} \\&\quad\left( p\ge 0 ,\, \mathfrak {R}(\kappa )>0, \, \mathfrak {R}(\mu )>0, \, m<\mathfrak {R}(\beta )< \mathfrak {R}(\gamma ), \,|z|<1\right) . \end{aligned}$$
(6)
The special case of (6) \(p=0,\,m=0\) is noted to reduce to the ordinary Gauss hypergeometric function \(F(\alpha ,\beta ;\gamma ;z)\) (Agarwal et al. 2015). Ibrahim et al. (2015b) introduced a generalized Noor integral operator using a fractional hypergeometric function as follows:
$$\begin{aligned}&Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m):\mathcal{A}\longrightarrow \mathcal{A}, \\&Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)\phi (z)= \Omega \left( zF^{a,b;\kappa ,\mu }_p(\alpha , \beta ;\gamma ;z;m)\right) ^{-1}*\phi (z),\,\,\,(z\in \mathcal{U}) \\&Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)\phi (z)=z+\sum _{n=2}^{\infty } \frac{(\gamma )_{n-1}}{(\alpha )_{n-1}(\beta )_{n-1}}\frac{\Omega \,B(\beta +n-1,\gamma -\beta +m)}{B^{a,b;\kappa ,\mu }_p(\beta +n-1,\gamma -\beta +m)}(\wp +1)_{n-1}\,a_n\,z^n, \end{aligned}$$
(7)
where
$$\Omega =\frac{B^{a,b;\kappa ,\mu }_p(\beta ,\gamma -\beta +m)}{B(\beta ,\gamma -\beta +m)},$$
$$\left( zF^{a,b;\kappa ,\mu }_p(\alpha ,\beta ;\gamma ;z;m)\right) ^{-1}= \sum _{n=1}^{\infty }\frac{(\gamma )_{n-1}}{(\alpha )_{n-1}(\beta )_{n-1}}\frac{B(\beta +n-1,\gamma -\beta +m)}{B^{a,b;\kappa ,\mu }_p(\beta +n-1,\gamma -\beta +m)}(\wp +1)_{n-1}\,z^n.$$
In view of (7), we get
$$z\left[ Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)\phi (z)\right] ^{\prime }=(\wp +1) \,Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)\phi (z)-\wp \,Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)\phi (z).$$
(8)
In particular, we have
$$Q^{\wp }_{0;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;0)\phi (z)= I_\wp (\alpha ,\beta ,\gamma )\phi (z)$$
where the integral operator \(I_\wp (\alpha ,\beta ,\gamma )\phi (z)\) was investigate by Noor (2006).

Making use of the principle of subordination various subordination theorems involving certain operators for analytic functions in \(\mathcal{U}\) were investigated by Miller and Mocanu (2000) and Owa and Srivastava (2004). Further, Bulboaca (2002a, b) and Miller and Mocanu (2003) extended the study to differential superordination as the dual problem of differential subordination, later the study has been taken by many researchers such as, Ali et al. (2005), Murugusundaramoorthy and Magesh (2006), and Shanmugam et al. (2006), Magesh and Murugusundaramoorthy (2008), Mostafa and Aouf (2009), Aouf and Mostafa (2010), Cho et al. (2011), Magesh (2011), Ibrahim et al. (2015a), and others.

Related to the present investigation, we mention some of them in recent years. In Ibrahim and Darus (2008), the first author applied a method based on the differential subordination and superordination in order to obtain results involving generalized Noor integral operator utilizing the Fox- Wright function for a normalized analytic function \(\phi (z),\, z\in \mathcal{U}\) and is denoted by \(I_\lambda [(\alpha _j,A_j)_{1,q};(\beta _j,B_j)_{1,p}]\phi (z)\). Also they studied the sufficient condition to satisfy
$$\eta _1(z)\prec \frac{[zI_\lambda [(\alpha _j, A_j)_{1,q};(\beta _j,B_j)_{1,p}]\phi (z)]'}{\Phi [I_\lambda [(\alpha _j, A_j)_{1,q};(\beta _j,B_j)_{1,p}]\phi (z)]}\prec \eta _2(z),$$
for some convex functions \(\eta _1\) and \(\eta _2\). with \(\eta _1(0)\ne 0\) and \(\eta _2(0)\ne 0\).
Aouf and Seoudy (2013) investigated some subordination and superordination results for certain p-valent functions in the open unit disc, which acted upon by a class of a linear operator denoted by \(I^{m,l}_{p,q,s,\lambda }(\alpha _1)\phi (z)\). Further, they studied the sufficient condition to satisfy
$$\eta _1(z)\prec \left[ \frac{I^{m,l}_{p,q,s,\lambda }(\alpha _1)\phi (z)}{z^p}\right] ^{\mu }\left[ \frac{z^p}{I^{m,l}_{p,q,s,\lambda }(\alpha _1+1)\phi (z)}\right] ^{\nu }\prec \eta _2(z),$$
for some convex functions \(\eta _1\) and \(\eta _2\). with \(\eta _1(0)\ne 0\) and \(\eta _2(0)\ne 0\).
Magesh et al. (2014) studied the subordination and superordination results of the linear operator denoted by \(\Theta [\alpha _1](\phi )(z)\). Also, they discussed the sufficient condition to satisfy
$$\eta _1(z)\prec \left[ \frac{\Theta [\alpha _1](\phi *\Phi )(z)}{z}\right] ^{\mu }\left[ \frac{z}{\Theta [\alpha _1+1](\phi *\Psi (z))}\right] ^{\nu }\prec \eta _2(z),$$
for some convex functions \(\eta _1\) and \(\eta _2\) with \(\eta _1(0)\ne 0\) and \(\eta _2(0)\ne 0\) and \(\Phi (z) = z + \sum _{n = 2}^\infty {\lambda _n z^n}, \Psi (z) = z + \sum _{n = 2}^\infty {\mu _n z^n}\) are analytic functions in \(\mathcal{U}\) with \(\lambda _n\ge 0, \mu _n\ge 0\,\, \text{ and }\,\, \lambda _n\ge \mu _n.\)
Ibrahim et al. (2015b) investigated some differential subordination and superordination results regarding the generalized integral operator defined by (7). Moreover, we investigate sufficient condition for a normalized analytic function \(\phi (z),z\in \mathcal{U}\) to satisfy
$$\eta _1(z)\prec \frac{z\left[ Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)\phi (z)\right] '}{\Phi \left[ Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)\phi (z)\right] }\prec \eta _2(z),$$
for some convex functions \(\eta _1\) and \(\eta _2\).
In this present paper, we study some differential subordination and superordination results for new subclasses regarding the generalized integral operator defined by (7). Moreover, we investigate sandwich results containing the given generalized integral operator for certain a normalized analytic function \(\phi (z),z\in \mathcal{U}\) such that \((\phi *\Theta )(z)\ne 0\) to satisfy
$$\eta _1(z)\prec \left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\prec \eta _2(z),$$
for some convex functions \(\eta _1\) and \(\eta _2\) and \(\Phi (z) = z + \sum _{n = 2}^\infty {\lambda _n z^n}, \Psi (z) = z + \sum _{n = 2}^\infty {\mu _n z^n}\) are analytic functions in \(\mathcal{U}\) with \(\lambda _n\ge 0,\mu _n\ge 0\,\, \text{ and }\,\, \lambda _n\ge \mu _n\).

Preliminaries

In order to prove our subordination and superordination results, we need to the following lemmas in the sequel.

Definition 1

(Miller and Mocanu 2003) Let \(\mathcal Q\) denote the set of functions \(\phi\) that are analytic and univalent on \(\overline{ \mathcal{U}}\backslash { \mathcal{E}(\phi )}\), where \({\mathcal{E}(\phi )}=\{\xi \in \partial \mathcal{U} :\lim \limits _{z\longrightarrow \xi }\phi (z)=\infty \}\), is such that \(\min |\phi '(\xi )|=p >0\) for \(\xi \in \partial { \mathcal{U}} \backslash {\mathcal{E}({\phi })}\).

Lemma 1

(Miller and Mocanu 2000) Let \(\rho\) be univalent in the unit disk \(\mathcal{U}\) and \(\Psi\) and let \(\Lambda\) be analytic in a domain \(\mathcal{D}\) containing \(\rho (\mathcal{U})\,\,\text{ with }\,\,\phi (\omega )\ne 0\) when \(\omega \in \eta (\mathcal{U})\). Set \(\mathcal{Q}(z):=z\eta ^{\prime }(z) \Lambda \left( \eta (z)\right) \,\,\text{ and }\,\,\vartheta (z) :=\Psi \left( \rho (z)\right) +\mathcal{Q}(z)\). Suppose that
  1. 1.

    \(\mathcal{Q}(z)\) is starlike univalent in \(\mathcal{U}\),

     
  2. 2.

    \(\mathfrak {R}(z\vartheta ^{\prime }(z)/\mathcal{Q}(z))>0\) for \(z \in \mathcal{U}\).

     
If \(\rho\) is analytic with \(\rho (0)=\eta (0),\,\rho (\mathcal{U})\subset \mathcal{D}\) and
$$\Psi \left( \rho (z)\right) +z\rho ^{\prime }(z)\Lambda \left( \rho (z)\right) \prec \Psi \left( \eta (z)\right) +z\eta ^{\prime }(z)\Lambda \left( \eta (z)\right) ,$$
then \(\rho (z)\prec \eta (z)\), and \(\eta (z)\) is the best dominant.

Lemma 2

(Bulboaca 2002a) Let \(\eta\) be convex univalent in the unit disk \(\mathcal{U}\) and \(\Pi\) and let \(\Delta\) be analytic in a domain \(\mathcal{D}\) containing \(\eta (\mathcal{U})\). Suppose that
  1. 1.

    \(z\eta ^{\prime }(z)\Delta \left( \eta (z)\right)\) is starlike univalent in \(\mathcal{U}\),

     
  2. 2.

    \(\mathfrak {R}\{\Pi '\left( \eta (z)\right) /\Delta (\eta (z))\})>0\) for \(z\in \mathcal{U}\).

     
If \(\rho (z)\in H[\eta (0),1]\cap \mathcal{Q}\) with \(\rho (\mathcal{U})\subseteq \mathcal{D}\) and \(\Pi \left( \rho (z)\right) +z\rho ^{\prime }(z)\Delta \left( \rho (z)\right)\) being univalent in \(\mathcal{U}\) and
$$\Pi \left( \eta (z)\right) +z\eta ^{\prime }(z)\Delta \left( \eta (z)\right) \prec \Pi \left( \rho (z)\right) +z\rho ^{\prime }(z)\Delta \left( \rho (z)\right) ,$$
then \(\eta (z)\prec \rho (z)\), and \(\eta (z)\) is the best subordinant.

Sandwich outcomes

By making use of Lemmas 1, we first prove the following subordination results.

Theorem 1

Let \(\Phi ,\Psi \in \mathcal{A},\,\lambda _i\in \mathbb {C}\,(i=1,2,3),\,\lambda _3\ne 0,\,\sigma ,\,\nu \in \mathbb {C}\) and \(\eta (z)\ne 0\) be univalent in \(\mathcal{U}\) such that \(z\eta ^{\prime }(z)/\eta (z)\) is starlike univalent in \(\mathcal{U}\) and
$$\mathfrak {R}\left\{ 1+\frac{\lambda _2}{\lambda _3}\,\eta (z)+\frac{z \eta ^{\prime \prime }(z)}{\eta ^{\prime }(z)}-\frac{z\eta ^{\prime }(z)}{\eta (z)}\right\} > 0, \quad (z \in \mathcal{U}).$$
(9)
If \(\phi \in \mathcal{A}\) satisfies the subordination
$$\begin{aligned} &\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}{z}\right] ^{\sigma } \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}\right] ^{\nu }\\&\quad+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] \\&\quad\prec \lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}, \end{aligned}$$
then
$$\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma } \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\prec \eta (z),$$
(10)
and \(\eta (z)\) is the best dominant.

Proof

Our aim is to apply Lemma 1. Setting
$$\rho (z)=\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}{z}\right] ^{\sigma } \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}\right] ^{\nu }.$$
Computation shows that
$$\begin{aligned} \frac{z\rho ^{\prime }(z)}{\rho (z)}&=\lambda _3\sigma (\wp +1) \left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] \\&\quad+\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu }(\gamma ;\alpha , \beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] , \end{aligned}$$
which yields the following subordination:
$$\lambda _1+\lambda _2\rho (z)+\lambda _3\,\frac{z\rho ^{\prime }(z)}{\rho (z)}\prec \lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}.$$
By setting
$$\Psi (\omega ):=\lambda _1+\lambda _2\,\omega ,\quad \Lambda (\omega ):=\frac{\lambda _3}{\omega },\,\,\,\pi \ne 0,$$
it can be easily observed that \(\Psi (\omega )\) is analytic in \(\mathbb {C}\) and \(\Lambda (\omega )\) is analytic in \(\mathbb {C}\backslash \{0\}\) and that \(\Lambda (\omega )\ne 0\) when \(\omega \in \mathbb {C}\backslash \{0\}\). Also, by letting
$$\begin{aligned} &\mathcal{Q}(z)=z\eta ^{\prime }(z)\Lambda (\eta (z))=\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)},\\&\vartheta (z)=\Psi \left( \eta (z)\right) +\mathcal{Q}(z)=\lambda _1+\lambda _2 \eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}, \end{aligned}$$
we find that \(\mathcal{Q}(z)\) is starlike univalent in \(\mathcal{U}\) and that
$$\mathfrak {R}\left\{ \frac{z\vartheta ^{\prime }(z)}{\mathcal{Q}(z)}\right\} =\mathfrak {R}\left\{ 1+\frac{\lambda _2}{\lambda _3}\,\eta (z)+\frac{z \eta ^{\prime \prime }(z)}{\eta ^{\prime }(z)}-\frac{z\eta ^{\prime }(z)}{\eta (z)}\right\} > 0.$$
\(\square\)

Corollary 1

Let the assumptions of Theorem 1 hold. Then the subordination
$$\begin{aligned}&\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] \\&\quad+\prec \lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}, \end{aligned}$$
implies
$$\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }\prec \eta (z),$$
and \(\eta (z)\) is the best dominant.

Proof

By letting \(\nu =0\) in Theorem 1, we have the required result. \(\square\)

Corollary 2

Let the assumptions of Theorem 1 hold. Then the subordination
$$\begin{aligned} &\lambda _1+\lambda _2\,\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)} \right] ^{\nu }+\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] \\&\quad\prec \lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}, \end{aligned}$$
implies
$$\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}\right] ^{\nu }\prec \eta (z),$$
and \(\eta (z)\) is the best dominant.

Proof

By letting \(\sigma =0\) in Theorem 1, we have the required result. \(\square\)

Theorem 2

Let \(\eta (z)\ne 0\) be convex univalent in the open unit disk \(\mathcal{U}\). Suppose that
$$\begin{aligned} \mathfrak {R}\left\{ \frac{\lambda _2}{\lambda _3}\,\eta (z)\right\} \ge 0,\quad \nu ,\,\pi \in \mathbb {C},\,\pi \ne 0,\,\,\text{ for }\,\,z \in \mathcal{U}, \end{aligned}$$
(11)
and that \(z\eta ^{\prime }(z)/\eta (z)\) is starlike univalent in \(\mathcal{U}\).
If \(\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\in H[1,1]\cap \mathcal{Q}\), where \(\Phi ,\,\Theta \in \mathcal{A},\)
$$\begin{aligned}&\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma } \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\\&\quad+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p; \kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] , \end{aligned}$$
is univalent in \(\mathcal{U}\) and the subordination
$$\begin{aligned} &\lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)} \\&\quad\prec\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi ) (z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\\&\quad+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha , \beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] , \end{aligned}$$
holds, then
$$\eta (z)\prec \left[ \frac{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu },$$
(12)
and \(\eta (z)\) is the best subordinant.

Proof

Our aim is to apply Lemma 2. Setting
$$\rho (z)=\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}{z}\right] ^{\sigma } \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}\right] ^{\nu }.$$
Computation shows that
$$\begin{aligned} \frac{z\rho ^{\prime }(z)}{\rho (z)}=\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] , \end{aligned}$$
which yields the following subordination:
$$\lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}\prec \lambda _1+\lambda _2\rho (z)+\lambda _3\,\frac{z\rho ^{\prime }(z)}{\rho (z)}.$$
By setting
$$\Pi (\omega ):=\lambda _1+\lambda +2\,\omega ,\quad \Delta (\omega ):=\frac{\lambda _3}{\omega },\,\,\,\pi \ne 0,$$
it can be easily observed that \(\Pi (\omega )\) is analytic in \(\mathbb {C}\) and \(\Delta (\omega )\) is analytic in \(\mathbb {C}\backslash \{0\}\) and that \(\Delta (\omega )\ne 0\) when \(\omega \in \mathbb {C}\backslash \{0\}\). Also, we obtain
$$\mathfrak {R}\left\{ \frac{\Pi '\left( \eta (z)\right) }{\Delta \left( \eta (z)\right) }\right\} =\mathfrak {R}\left\{ \frac{\lambda _2}{\lambda _3}\,\eta (z)\right\} > 0.$$
Then the relation (12) follows by an application of Lemma 2. \(\square\)

Corollary 1

Let the assumptions of Theorem 1 hold. Then the subordination
$$\begin{aligned}&\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi ) (z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] \\&\quad+\prec \lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}, \end{aligned}$$
implies
$$\eta (z)\prec \left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Phi )(z)}{z}\right] ^{\sigma },$$
and \(\eta (z)\) is the best subordinant.

Proof

By letting \(\nu =0\) in Theorem 1, we have the required result. \(\square\)

Corollary 3

Let the assumptions of Theorem 1 hold. Then the subordination
$$\begin{aligned}&\lambda _1+\lambda _2\,\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }+\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] \\&\quad\prec \lambda _1+\lambda _2\eta (z)+\lambda _3\,\frac{z\eta ^{\prime }(z)}{\eta (z)}, \end{aligned}$$
implies
$$\eta (z)\prec \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu },$$
and \(\eta (z)\) is the best subordinant.

Proof

By letting \(\sigma =0\) in Theorem 1, we have the required result. \(\square\)

Combining Theorems 1 and 2, in order to get the following sandwich theorem.

Theorem 3

Let \(\eta _1(z)\ne 0,\,\,\eta _2(z)\ne 0\) be convex univalent in the open unit disk \(\mathcal{U},\,\lambda _i\in \mathbb {C}\,(i=1,2,3), \,\lambda _3\ne 0,\,\sigma ,\,\nu \in \mathbb {C}\) and let \(\eta _2(z)\) satisfy (9) and \(\eta _1(z)\) satisfy (11) respectively. Suppose that and that \(z\eta _i'(z)/\eta _i(z),\,\,i=1,2\), is starlike univalent in \(\mathcal{U}\). If \(\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma } \left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\in H[1,1]\cap \mathcal{Q}\), where \(\Phi ,\,\Theta \in \mathcal{A}\),
$$\begin{aligned}&\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ; \alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}\right] ^{\nu }\\&\quad+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m) (\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] , \end{aligned}$$
is univalent in \(\mathcal{U}\) and the subordination
$$\begin{aligned}&\lambda _1+\lambda _2\eta _1(z)+\lambda _3\,\frac{z\eta ^{\prime }_1(z)}{\eta _1(z)} \\&\quad\prec\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\\&\quad+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] \\&\quad\prec \lambda _1+\lambda _2\eta _2(z)+\lambda _3\,\frac{z\eta ^{\prime }_2(z)}{\eta _2(z)}, \end{aligned}$$
holds, then
$$\eta _1(z)\prec \left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\prec \eta _2(z),$$
and \(\eta _1(z)\) is the best subordinant and \(\eta _2(z)\) is the best dominant. \(\square\)

Corollary 4

Let the assumptions of Theorem 3 hold and satisfy the subordination
$$\begin{aligned}&\lambda _1+\lambda _2\frac{1+A_1z}{1+B_1z}+\lambda _3\frac{(A_1-B_1)z}{(1+A_1z)(1+B_1z)}\, \\&\quad\prec\lambda _1+\lambda _2\,\left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z}\right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\\&\quad+\lambda _3\sigma (\wp +1)\left[ \frac{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}-1\right] +\nu (\wp +2)\left[ 1-\frac{Q^{\wp +2}_{p;\kappa ,\mu } (\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}{Q^{\wp +1}_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] \\&\quad\prec \lambda _1+\lambda _2\frac{1+A_2z}{1+B_2z}+\lambda _3\,\frac{(A_2-B_2)z}{(1+A_2z)(1+B_2z)}, \end{aligned}$$
Then
$$\frac{1+A_1z}{1+B_1z}\prec \left[ \frac{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Phi )(z)}{z} \right] ^{\sigma }\left[ \frac{z}{Q^{\wp }_{p;\kappa ,\mu }(\gamma ;\alpha ,\beta ;z;m)(\phi *\Theta )(z)}\right] ^{\nu }\prec \frac{1+A_2z}{1+B_2z},$$
and \(\frac{1+A_1z}{1+B_1z}\) is the best subordinant and \(\frac{1+A_2z}{1+B_2z}\) is the best dominant.

Proof

Setting \(\eta _1(z)=\frac{1+A_1z}{1+B_1z}\,(-1\le B_1<A_1\le 1)\) and \(\eta _2(z)=\frac{1+A_2z}{1+B_2z}\,(-1\le B_2 <A_2\le 1)\) in Theorem 3. \(\square\)

Conclusion

By the term of the extend fractional hypergeometric function, we defined a new fractional integral operator in the open unit disk. This operator is a generalization of the well known Noor integral operator. Based on this operator, many subclasses may introduce in the geometric function theory. In this study, we concerned with a specific class of analytic function called of convolution type. This class imposed several well known classes.

Declarations

Authors’ contributions

RWI, MZA and HFA jointly worked on deriving the results. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Faculty of Computer Science and Information Technology, University Malaya
(2)
Institute of Engineering Mathematics, Universiti Malaysia Perlis

References

  1. Agarwal P, Choi J, Paris RB (2015) Extended Riemann–Liouville fractional derivative operator and its applications. J Nonlinear Sci Appl 8:451–466Google Scholar
  2. Ali R, Ravichandran V, Khan MH, Subramanian G (2005) Differential sandwich theorems for certain analytic functions. Far East J Math Sci 15:87–94Google Scholar
  3. Aouf MK, Mostafa AO (2010) Theorems for analytic functions defined by convolution. Acta Univ Apulensis Math Inform 21:7–20Google Scholar
  4. Aouf MK, Seoudy TM (2013) Differential sandwich theorems of p-valent analtic functions involving a linear operator. An Şt Univ Ovidius Constanţa 21(1):5–18Google Scholar
  5. Bulboaca T (2002a) A class of superordination-preserving integral operators. Indag Math (NS) 13:301–311View ArticleGoogle Scholar
  6. Bulboaca T (2002b) Classes of first-order differential superordinations. Demonstr Math 35(2):287–292Google Scholar
  7. Cho NE, Kwon OS, Ali R, Ravichandaran V (2011) Subordination and superordination for multivalent functions associated with the Dziok–Srivastava operator. J Inequal Appl. Article ID 486595Google Scholar
  8. de Branges L (1985) A proof of the Bieberbach conjecture. Acta Math 154(1–2):137–152View ArticleGoogle Scholar
  9. Ibrahim RW, Ahmad MZ, Al-Janaby HF (2015a) Third-order differential subordination and superordination involving a fractional operator. Open Math 13(1):706–728. doi:https://doi.org/10.1515/math-2015-0068 Google Scholar
  10. Ibrahim RW, Ahmad MZ, Al-Janaby HF (2015b) Upper and lower bounds of integral operator defined by the fractional hypergeometric function. Open Math 13(1):768–780. doi:https://doi.org/10.1515/math-2015-0071 Google Scholar
  11. Ibrahim RW, Darus M (2008) New classes of analytic functions involving generalized Noor integral operator. J Inequal Appl. Article ID 390435, pp 1–14Google Scholar
  12. Magesh N (2011) Differential sandwich results for certain subclasses of analytic functions. Math Comput Model 54:803–814View ArticleGoogle Scholar
  13. Magesh N, Jothibasu J, Murthy S (2014) Subordination and superordination results associated with the generalized hypergeometric function. Math Slovaca 64(5):1197–1216View ArticleGoogle Scholar
  14. Magesh N, Murugusundaramoorthy G (2008) Differential subordinations and superordinations for a comprehensive class of analytic functions. SUT J Math 44:237–255Google Scholar
  15. Miller SS, Mocanu PT (2000) Differential subordinations, theory and applications, monographs and textbooks in pure and applied mathematics, vol 225. Dekker, New YorkGoogle Scholar
  16. Miller SS, Mocanu PT (2003) Subordinants of differetial superordinations. Complex Var Theory Appl 48(10):815–826View ArticleGoogle Scholar
  17. Mostafa AO, Aouf MK (2009) Sandwich theorems for certain subclasses of analytic functions defined by family of linear operators. J Appl Anal 15:269–280View ArticleGoogle Scholar
  18. Murugusundaramoorthy G, Magesh N (2006) Differential subordinations and superordinations for analytic functions defined by the Dziok–Srivastava linear operator. J Inequal Pure Appl Math 7:1–9Google Scholar
  19. Noor KL (2006) Integral operators defined by convolution with hypergeometric functions. Appl Math Comput 182:1872–1881Google Scholar
  20. Owa S, Srivastava HM (2004) Some subordination theorems involving a certain family of integral operators. Integral Transform Spec Funct 15:445–454View ArticleGoogle Scholar
  21. Shanmugam TN, Ravichandran V, Sivasubramanian S (2006) Differential sandwich theorems for some subclasses of analytic functions. Aust J Math Anal Appl 3(1):1–11Google Scholar
  22. Srivastava HM, Agarwal P, Jain S (2014) Generating functions for the generalized Gauss hypergeometric functions. Appl Math Comput 247:348–352Google Scholar
  23. Srivastava HM, Choi J (2012) Zeta and q-zeta functions and associated series and integrals. Elsevier, AmsterdamGoogle Scholar

Copyright

© Ibrahim et al. 2016