Open Access

More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings

SpringerPlus20165:77

https://doi.org/10.1186/s40064-016-1683-x

Received: 27 August 2015

Accepted: 6 January 2016

Published: 26 January 2016

Abstract

Our aim in this article is to incorporate the notion of “strongly s-convex function” and prove a new integral identity. Some new inequalities of Simpson type for strongly s-convex function utilizing integral identity and Holder’s inequality are considered.

Keywords

Simpson’s inequality Strongly s-convex function Integral identity Holder’s integral inequality

Mathematics Subject Classification

26D15 26E60 41A55

Background

The following definition is well known in the literature as convex function:

Let \(f:I\subseteq R\rightarrow R\) be a function defined on the interval I of real numbers. Then f is called convex, if \(f\left( {\lambda x+\left( {1-\lambda } \right) y} \right) \le \lambda f\left( x \right) +\left( {1-\lambda } \right) f\left( y \right) ,\) for all \(x,y\in I\) and \(\lambda \in \left[ {0,1} \right] .\) Geometrically, this means that if P, Q and R are three distinct points on graph of f with Q between P and R, then Q is on or below chord PR.

Hudzik and Maligrada (1994) defined s-convex function as: A function \(f:\left[ {0,\infty } \right) \rightarrow R\) is said to be s-convex function in the first sense, if \(f\left( {\mu x+\nu y} \right) \le \mu ^{s} f\left( x \right) +\nu ^{s} f\left( y \right)\), for all \(x,y\in \left[ {0,\infty } \right)\) and \(\mu ,\nu \ge 0,\) with \(\mu ^{s} + \nu ^{s} = 1\). We denote this by \(K_\alpha ^1\), for some fixed \(s \in (0, 1]\). Also \(f:\left[ {0,\infty } \right) \rightarrow R\) is said to be s-convex function in the second sense, if above condition holds for all \(x,y\in \left[ {0,\infty } \right)\) and \(\mu ,\nu \ge 0,\) with \(\mu + \nu = 1\).

The following is very important and well known in the literature, as Simpson’s inequality:
$$\begin{aligned} \left| {\int \limits _a^b {f(x)dx} -\frac{(b-a)}{3}\left[ {\frac{f(a)+f(b)}{2}+2f\left( {\frac{a+b}{2}} \right) } \right] } \right| \le \frac{1}{2880}\left\| {f^{\left( 4 \right) }} \right\| _\infty \left( {b-a} \right) ^5, \end{aligned}$$
(1)
where the mapping \(f:[a,b]\rightarrow R\) is supposed to be a four times continuously differentiable on the interval \(\left( {a,b} \right)\) and having the fourth derivative bounded on \(\left( {a,b} \right) ,\) that is
$$\begin{aligned} \left\| {f^{\left( 4 \right) }} \right\| _\infty =\sup { }_{x\in \left( {a,b} \right) }\left| {f^{\left( 4 \right) }\left( x \right) } \right| <\infty . \end{aligned}$$
Dragomir et al. (2000) proved that: Let \(f:[a, b] \rightarrow R\) be a differentiable function on \(I^0\) (interior of I) \(a,b\in I\) with \(a<b.\) If the mapping \(\left| {{f}'} \right|\) is convex on \(\left[ {a,b} \right] ,\) then we have the following inequality:
$$\begin{aligned} \left| {\frac{f\left( a \right) +f\left( b \right) }{2}-\frac{1}{b-a}\int _a^b {f\left( x \right) dx} } \right| \le \frac{\left( {b-a} \right) \left( {\left| {{f}'\left( a \right) } \right| +\left| {{f}'\left( b \right) } \right| } \right) }{8}. \end{aligned}$$
(2)
Sarikaya et al. (2010) showed that: Let \(f: [a, b]\rightarrow R\) be a differentiable function on \(I^0\)(interior of I) such that \(f^{'} \in L_{1}[a, b]\), where \(a,b\in I\) with \(a<b.\) If the mapping \(\left| {{f}'} \right|\) is s-convex on \(\left[ {a,b} \right] ,\) for some fixed \(s\in \left( {0,1} \right] ,\) then we have the following inequality
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f(a)+4f\left( {\frac{a+b}{2}} \right) +f(b)} \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \nonumber \\&\quad \le \frac{\left( {s-4} \right) 6^{s+1}+2\times 5^{s+2}-2\times 3^{s+2}+2}{6^{s+2}\left( {s+1} \right) \left( {s+2} \right) }\left( {b-a} \right) \left( {\left| {{f}'\left( a \right) } \right| +\left| {{f}'\left( b \right) } \right| } \right) . \end{aligned}$$
(3)
Alomari et al. (2011) established that: Let \(f:I\subset R\rightarrow R\) be a differentiable function on \(I^0\) (interior of I)\(a,b\in I\) with \(a<b.\) If the mapping \(\left| {{f}'} \right|\) is s-convex on \(\left[ {a,b} \right] ,\) for some \(s\in \left( {0,1} \right] ,\) then we have the following inequality:
$$\begin{aligned}&\left| {f\left( {\frac{a+b}{2}} \right) -\frac{1}{b-a}\int _a^b {f\left( x \right) dx} } \right| \nonumber \\&\quad \le \frac{b-a}{4\left( {s+1} \right) \left( {s+2} \right) }\left\{ {\left| {{f}'\left( a \right) } \right| +\left| {{f}'\left( b \right) } \right| +2\left( {s+1} \right) \left| {f^{'}\left( {\frac{a+b}{2}} \right) } \right| } \right\} \nonumber \\&\quad \le \frac{\left( {2^{2-s}+1} \right) \left( {b-a} \right) }{4\left( {s+1} \right) \left( {s+2} \right) }\left[ {\left| {{f}'\left( a \right) } \right| +\left| {{f}'\left( b \right) } \right| } \right] . \end{aligned}$$
(4)
For utilizing different kinds of convexities, additional findings relating to the Simpson’s type inequality, readers are directed to Dragomir et al. (2000), Qaisar et al. (2013), Hussain and Qaisar (2014), Dragomir (1999), Wang et al. (2013), Xi and Qi (2013) and Pearce and Pecari’c (2000).

Main results

To prove our main result, we need the following definition and lemma.

Definition 1

(Polyak 1996) Let \(f:I\subseteq R\rightarrow R\) is said to be strongly s-convex with modulus \(c>0\) and for some fixed \(s\in \left( {0,1} \right]\), if
$$\begin{aligned} f\left( {\lambda x+\left( {1-\lambda } \right) y} \right) \le \lambda ^{s} f\left( x \right) +\left( {1-\lambda } \right) ^{s}f\left( y \right) -c\lambda \left( {1-\lambda } \right) \left( {x-y} \right) ^2, \end{aligned}$$
for all \(x,y\in I\) and \(\lambda \in \left[ {0,1} \right] .\)

Observation 2

It is clear that, any strongly s-convex function is a strong convex function but the converse is not true in general.

Now we prove the following lemma:

Lemma 3

Let \(f: I = \left[ {a,b} \right] \subset R\rightarrow R\) be such that \({f}'\) is absolutely continuous and \({f}''\in L_1 \left( {\left[ {a,b} \right] } \right) .\) Then the following inequality holds:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \le \frac{\left( {b-a} \right) ^2}{96} \nonumber \\&\quad \times \int \limits _0^1 {\psi \left( {1-\psi } \right) \left[ {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) +{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) +{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right] } d\psi . \end{aligned}$$
(5)

Proof

Using integrating by parts, we have
$$\begin{aligned}&\int \limits _0^1 {\psi \left( {1-\psi } \right) {f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) d\psi } \\&\quad =-\frac{4}{b-a}\left[ {\psi \left( {1-\psi } \right) \left. {{f}'\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| _0^1 -\int \limits _0^1 {\left( {1-2\psi } \right) {f}'\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) d\psi } } \right] \\&\quad =-\frac{16}{\left( {b-a} \right) ^2}\left[ {\left( {1-2\psi } \right) \left. {f\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| _0^1 +2\int \limits _0^1 {f\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) d\psi } } \right] \\&\quad =\frac{16}{\left( {b-a} \right) ^2}\left[ {f\left( a \right) +f\left( {\frac{3a+b}{4}} \right) -\frac{96}{\left( {b-a} \right) ^3}\int \limits _a^{{\left( {3a+b} \right) } /4} {f\left( x \right) dx} } \right] . \\ \end{aligned}$$
Analogously,
$$\begin{aligned}&\int \limits _0^1 {\psi \left( {1-\psi } \right) {f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) d\psi } \\&\quad =\frac{16}{\left( {b-a} \right) ^2}\left[ {f\left( {\frac{3a+b}{4}} \right) +f\left( {\frac{a+3b}{4}} \right) } \right] -\frac{96}{\left( {b-a} \right) ^3}\int \limits _{{\left( {3a+b} \right) } / 4}^{{\left( {a+3b} \right) }/4} {f\left( x \right) dx}. \\ \end{aligned}$$
And
$$\begin{aligned}&\int \limits _0^1 {\psi \left( {1-\psi } \right) {f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) d\psi } \\&\quad =\frac{16}{\left( {b-a} \right) ^2}\left[ {f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{96}{\left( {b-a} \right) ^3}\int \limits _{{\left( {a+3b} \right) } /4}^b {f\left( x \right) dx}. \end{aligned}$$
This proves as required.

Theorem 4

Let \(f: I = \left[ {a,b} \right] \subset R\rightarrow R\) be such that \({f}'\) is absolutely continuous and \({f}''\in L_1 \left( {\left[ {a,b} \right] } \right) .\) If the mapping \(\left| {{f}''} \right|\) is strongly s-convex on \(\left[ {a,b} \right] ,\) for \(q\ge 1\) and for some fixed \(s\in \left( {0,1} \right]\) , then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{6^{1 / q}\left( {b-a} \right) ^2}{576}\left\{ {\begin{array}{l} \left[ {\begin{array}{l} \frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{17c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 /q} \\ +\left[ {\begin{array}{l} \frac{\left( {s -1} \right) 2^{s +2}+s +5}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -3} \right) 3^{s +2}+\left( {s +7} \right) 2^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{37c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 / q} \\ +\left[ {\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q-\frac{17c\left( {b-a} \right) ^2}{960}} \right] ^{1/q} \end{array}} \right\} .\\ \end{aligned}$$

Proof

Using Lemma 3 and strongly s-convexity of \(\left| {{f}''} \right| ^q\), we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) } d\psi } \right] ^{1-1 / q}\left\{ {\begin{array}{l} \left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {\begin{array}{l} \left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi } \right) ^2\left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } } \right] ^{1 /q} \\ +\left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {\begin{array}{l} \left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi ^2} \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } } \right] ^{1 / q} \\ +\left[ {\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {\begin{array}{l} \left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi ^2\left( {1-\psi } \right) \left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } } \right] ^{1 /q} \\ \end{array}} \right\} \\&\quad =\frac{6^{1 / q}\left( {b-a} \right) ^2}{576}\left\{ {\begin{array}{l} \left[ {\begin{array}{l} \frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{17c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 /q} \\ +\left[ {\begin{array}{l} \frac{\left( {s -1} \right) 2^{s +2}+s +5}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -3} \right) 3^{s +2}+\left( {s +7} \right) 2^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q \\ -\frac{37c\left( {b-a} \right) ^2}{960} \\ \end{array}} \right] ^{1 /q} \\ +\left[ {\frac{1}{\left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( a \right) } \right| ^q+\frac{\left( {s -5} \right) 4^{s +2}+\left( {s +9} \right) 3^{s +2}}{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left| {{f}''\left( b \right) } \right| ^q-\frac{17c\left( {b-a} \right) ^2}{960}} \right] ^{1 / q} \\ \end{array}} \right\} . \end{aligned}$$

Corollary 5

Under the conditions of Theorem  4,
  1. 1.
    If \(q = 1\), then
    $$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {s -3} \right) 4^{s +2}+\left( {2s +6} \right) \left( {3^{s +2}+1} \right) }{\left( {s +1} \right) \left( {s +2} \right) \left( {s +3} \right) 4^s }\left( {\left| {{f}''\left( a \right) } \right| +\left| {{f}''\left( b \right) } \right| } \right) -\frac{71c\left( {b-a} \right) ^2}{960}} \right] . \end{aligned}$$
     
  2. 2.
    If \(q = 1\) and \(s = 1\), then
    $$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\left( {\left| {{f}''\left( a \right) } \right| +\left| {{f}''\left( b \right) } \right| } \right) -\frac{71c\left( {b-a} \right) ^2}{960}} \right] . \end{aligned}$$
     

Theorem 6

Let f be defined as in Theorem 4 and the mapping \(\left| {{f}''} \right| ^q\) is strongly s-convex on \(\left[ {a,b} \right] ,\) for \(q>1\) and for some fixed \(s\in \left( {0,1} \right]\) , then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {B\left( {\frac{2q-1}{q-1},\frac{2q-1}{q-1}} \right) } \right] ^{1-1 /q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) }} \right) } \right] ^{1/q} \\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1/ q} \\ +\left[ {\left( {2^{s +1}-1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}-2^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{11c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \\ \end{array}} \right\} , \end{aligned}$$
where \(B\left( {\alpha ,\beta } \right)\) is the classical Beta function which may be defined by
$$\begin{aligned} B\left( {\alpha ,\beta } \right) =\int \limits _0^1 {\psi ^{\alpha -1}\left( {1-\psi } \right) ^{\beta -1}} d\psi ,\quad s ,\beta >0. \end{aligned}$$

Proof

Using Lemma 3, strong s-convexity of \(\left| {{f}''} \right| ^q\) and Holder’s inequality, we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \begin{array}{l} \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\left[ {\psi \left( {1-\psi } \right) } \right] } ^{q / {\left( {q-1} \right) }}d\psi } \right] ^{1-1 / q}\left\{ {\begin{array}{l} \left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {\left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi } \right) \left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 /q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {\left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi } \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 /q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {\left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1/q} \\ \end{array}} \right\} \\ \end{array}\\&\qquad \times \begin{array}{l} \left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\ \le \frac{\left( {b-a} \right) ^2}{96}\left[ {B\left( {\frac{2q-1}{q-1},\frac{2q-1}{q-1}} \right) } \right] ^{1-1/q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) }} \right) } \right] ^{1/q} \\ \end{array}\\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1/q} \\ +\left[ {\left( {2^{s +1}-1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}-2^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{11c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +1}-3^{s +1}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{5c\left( {b-a} \right) ^2\left( {s +1} \right) 4^s }{48}} \right] ^{1 /q} \end{array}} \right\} . \end{aligned}$$
This completes the proof.

Theorem 7

Let f be defined as in Theorem  4 and the mapping \(\left| {{f}''} \right| ^q\) is strongly s-convex on \(\left[ {a,b} \right] ,\) for \(q>1\) and for some fixed \(s\in \left( {0,1} \right]\) , then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1 / q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1/ q} \\&\qquad \times \,\left\{ {\begin{array}{l} \left[ {\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ +\left[ {\left( {2^{s +1}s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +2}-2^{s +1}\left( {s +4} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{23c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ +\left[ {\left( {s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +2}-3^{s +1}\left( {s +5} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ \end{array}} \right\} . \\ \end{aligned}$$

Proof

Using Lemma 3, Holder’s inequality and strongly s- convexity of \(\left| {{f}''} \right| ^q\), we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\left[ {\psi \left( {1-\psi } \right) } \right] } ^{q / {\left( {q-1} \right) }}d\psi } \right] ^{1-1/ q}\left\{ {\begin{array}{l} \left[ {\left( {\begin{array}{l} \int \limits _0^1 {\psi \left( {\left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\psi \left( {\left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1+\psi } \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\psi \left( {\left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi ^2\left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ \end{array}} \right\} . \\ \end{aligned}$$
or
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1 / q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1/q} \\&\quad \times \, \left\{ {\begin{array}{l} \left[ {\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ +\left[ {\left( {2^{s +1}s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +2}-2^{s +1}\left( {s +4} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{23c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ +\left[ {\left( {s +1} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {4^{s +2}-3^{s +1}\left( {s +5} \right) } \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ \end{array}} \right\} \end{aligned}$$
This completes the proof.

Theorem 8

Let f be defined as in Theorem 4 and the mapping \(\left| {{f}''} \right| ^q\) is strongly s-convex on \(\left[ {a,b} \right] ,\) for \(q>1\) and for some fixed \(s\in \left( {0,1} \right]\) , then we have the following inequality:
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1/ q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1 /q} \\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +2}-\left( {s +5} \right) 3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {s +1} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/q} \\ +\left[ {\left( {2^{s +2}-s -3} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}\left( {s -1} \right) -2^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{21c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ \end{array}} \right\} . \end{aligned}$$

Proof

Using Lemma 3, Holder inequality and strongly s- convexity of \(\left| {{f}''} \right|\), we have
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\begin{array}{l} \int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{3+\psi }{4}a+\frac{1-\psi }{4}b} \right) } \right| } d\psi \\ +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{1+\psi }{4}a+\frac{3-\psi }{4}b} \right) } \right| } d\psi +\int \limits _0^1 {\psi \left( {1-\psi } \right) \left| {{f}''\left( {\frac{\psi }{4}a+\frac{4-\psi }{4}b} \right) } \right| } d\psi \\ \end{array}} \right] \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\int \limits _0^1 {\left( {1-\psi } \right) } \psi ^{q /{\left( {q-1} \right) }}d\psi } \right] ^{1-1 / q}\left\{ {\begin{array}{l} \left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {1-\psi } \right) \left( {\left( {\frac{3+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{1-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi } \right) ^2\left( {3+\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {1-\psi } \right) \left( {\left( {\frac{1+\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{3-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\left( {1-\psi ^2} \right) \left( {3-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ +\left[ {\left( {\begin{array}{l} \int \limits _0^1 {\left( {1-\psi } \right) \left( {\left( {\frac{\psi }{4}} \right) ^s \left| {{f}''\left( a \right) } \right| ^q+\left( {\frac{4-\psi }{4}} \right) ^s \left| {{f}''\left( b \right) } \right| ^q} \right) } \\ -\frac{c\left( {b-a} \right) ^2}{16}\int \limits _0^1 {\psi \left( {1-\psi } \right) \left( {4-\psi } \right) d\psi } \\ \end{array}} \right) } \right] ^{1 / q} \\ \end{array}} \right\} . \\ \end{aligned}$$
or
$$\begin{aligned}&\left| {\frac{1}{6}\left[ {f\left( a \right) +2f\left( {\frac{3a+b}{4}} \right) +2f\left( {\frac{a+3b}{4}} \right) +f\left( b \right) } \right] -\frac{1}{b-a}\int \limits _a^b {f(x)dx} } \right| \\&\quad \le \frac{\left( {b-a} \right) ^2}{96}\left[ {\frac{\left( {q-1} \right) ^2}{\left( {2q-1} \right) \left( {3q-2} \right) }} \right] ^{1-1/ q}\left[ {\left( {\frac{1}{4^s \left( {s +1} \right) \left( {s +2} \right) }} \right) } \right] ^{1/ q} \\&\qquad \times \, \left\{ {\begin{array}{l} \left[ {\left( {4^{s +2}-\left( {s +5} \right) 3^{s +1}} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {s +1} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{13c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 / q} \\ +\left[ {\left( {2^{s +2}-s -3} \right) \left| {{f}''\left( a \right) } \right| ^q+\left( {3^{s +1}\left( {s -1} \right) -2^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{21c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1/ q} \\ +\left[ {\left| {{f}''\left( a \right) } \right| ^q+\left( {\left( {s -2} \right) 4^{s +1}+3^{s +2}} \right) \left| {{f}''\left( b \right) } \right| ^q-\frac{7c\left( {b-a} \right) ^2\left( {s +1} \right) \left( {s +2} \right) 4^s }{192}} \right] ^{1 /q} \\ \end{array}} \right\} . \end{aligned}$$
This completes the proof. \(\square\)

Conclusion

We incorporated notion of “strongly s-convex function” and proved a new integral identity. Some new inequalities of Simpson type for strongly s-convex function utilizing integral identity and Holder’s inequality are also considered. These results give better estimates as presented earlier in the literature.

Declarations

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgements

There authors are thankful to the reviewers for their fruitful comments towards the improvement of the paper.

Competing interest

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)
Department of Mathematics, College of Science, Qassim University
(2)
Department of Mathematics, COMSATS Institute of Information Technology

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Copyright

© Hussain and Qaisar. 2016