PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114)) (2016), 251–257 DOI: 10.2298/PIM1614251N

HERMITE–HADAMARD INEQUALITIES FOR DIFFERENTIABLE p-CONVEX FUNCTIONS USING HYPERGEOMETRIC FUNCTIONS

Muhammad Aslam Noor, Muhammad Uzair Awan, Marcela V. Mihai, and Khalida Inayat Noor

Abstract. We derive some new integral identities for differentiable functions. Then using these auxiliary results, we obtain new Hermite–Hadamard type inequalities for differentiable p-convex functions. Some special cases are also discussed.

1. Introduction

Recently theory of convexity has received much attentions by many researchers. Consequently the classical concepts of convex sets and convex functions have been extended and generalized in several directions using novel and innovative ideas, see [1]. Zhang [11] introduced the notion of p-convex functions. It is worth to mention here that besides the classical convex functions, the class of p-convex functions also includes the class of harmonically convex functions introduced and studied by Iscan [5]. For some recent investigations on p-convex functions, see [4].

The interrelationship between theory of convex functions and theory of inequalities has attracted many researchers. One of the most extensively studied inequality for convex functions is the Hermite–Hadamard inequality. This inequality provides the necessary and sufficient condition for a function to be convex. For some recent investigation on Hermite–Hadamard type inequalities, see [2–10].

In this article, We consider the class of p-convex functions. We derive two new integral identities for differentiable functions. Using these results we establish our main results that are Hermite–Hadamard type inequalities for differentiable p-convex functions. We use hypergeometric functions to solve our integrals. It is expected that the ideas and techniques of this paper may stimulate further research in this area. This is the main motivation of this paper.

2010 Mathematics Subject Classification: 26D15; 26A51.

Key words and phrases: convex functions; p-convex functions; Hermite–Hadamard inequalities.

Communicated by Gradimir Milovanović.

2. Preliminaries and lemmas

In this section, we recall some previously known concepts and derive some new results which play an important role in the development of our main results.

Definition 2.1. [11] An interval I is said to be a p-convex set, if

M_{p}(x, y; t) = [tx^{p} + (1 − t)y^{p}]^{\frac{1}{p}} ∈ I,

for all x, y ∈ I, t ∈ [0, 1], where p = 2k + 1 or p = \frac{n}{m} , n = 2r + 1, m = 2t + 1 and k, r, t ∈ ℕ.

Definition 2.2. [11] Let I be a p-convex set. A function f: I → ℝ is said to be p-convex function or belongs to the class P C(I), if

f(M_{p}(x, y; t)) ⩽ tf(x) + (1 − t)f(y), ∀x, y ∈ I, t ∈ [0, 1].

It is obvious that for p = 1, Definition 2.2 reduces to the definition for classical convex functions. Note that for p = −1, we have the definition of harmonically convex functions.

Definition 2.3. [5] A function f: I ⊂ ℝ∖{0} → ℝ is said to be harmonically convex function, if

f( \frac{xy}{(1 − t)x + ty} ) ⩽ tf(x) + (1 − t)f(y), ∀x, y ∈ I, t ∈ [0, 1].

Also note that for t = \frac{1}{2} in Definition 2.2, we have Jensen p-convex functions or mid p-convex functions

f(M_{p}(x, y; 1/2)) ⩽ \frac{f(x) + f(y)}{2} , ∀x, y ∈ I, t ∈ [0, 1].

Now we derive some new integral identities; I^{0} will denote the interior of I.

Lemma 2.4. Let f: I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0} with a < b. If f′ ∈ L[a, b], then

\frac{f(a) + f(b)}{2} − \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx

= \frac{b^{p} − a^{p}}{2p} ∫_{0}^{1} M_{p}^{−1} (a, b; t)(1 − 2t)f′(M_{p}(a, b; t)) dt,

where M_{p}^{−1} (a, b; t) = [ta^{p} + (1 − t)b^{p}]^{\frac{1}{p} −1} .

Proof. It suffices to show that

∫_{0}^{1} M_{p}^{−1} (a, b; t)(1 − 2t)f′(M_{p}(a, b; t)) dt
= \frac{f(a) + f(b)}{b^{p} − a^{p}} − \frac{2p}{b^{p} − a^{p}} ∫_{0}^{1} f([ta^{p} + (1 − t)b^{p}]^{\frac{1}{p}} ) dt
= \frac{f(a) + f(b)}{b^{p} − a^{p}} − \frac{2p^{2}}{(b^{p} − a^{p})^{2}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx.

INEQUALITIES FOR DIFFERENTIABLE p-CONVEX FUNCTIONS 253

Multiplying both sides of the above inequality by \frac{b^{p} −a^{p}}{2p} , we get the required result. □

Note that for p = 1, Lemma 2.4 reduces to the following known integral identity by Dragomir et al. [2].

Lemma 2.5. [2] Let f: I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0} with a < b. If f′ ∈ L[a, b], then, we have

\frac{f(a) + f(b)}{2} − \frac{1}{b − a} ∫_{a}^{b}f(x) dx = \frac{b − a}{2} ∫_{0}^{1}(1 − 2t)f′(ta + (1 − t)b) dt.

If p = −1, then Lemma 2.4 reduces to the following integral identity mainly due to Iscan [5].

Lemma 2.6. [5] Let f : I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0} with a < b. If f′ ∈ L[a, b], then

\frac{f(a) + f(b)}{2} − \frac{ab}{b − a} ∫_{a}^{b} \frac{f(x)}{x^{2}} dx

= \frac{ab(b − a)}{2} ∫_{0}^{1} \frac{1 − 2t}{[tb + (1 − t)a]^{2}} f′( \frac{ab}{tb + (1 − t)a} ) dt.

Lemma 2.7. Let f: I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0} with a < b. If f′ ∈ L[a, b], then

\frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx − f( [ \frac{a^{p} + b^{p}}{2} ]^{\frac{1}{p}} )

= \frac{b^{p} − a^{p}}{p} ∫_{0}^{1} M_{p}^{−1} (a, b; t)ϑ(t)f′(M_{p}(a, b; t)) dt,

where

ϑ(t) = \{ \begin{matrix}t, & [0, \frac{1}{2}), \\ t − 1, & [\frac{1}{2}, 1]. \end{matrix}

Proof. A simple integration by parts completes the proof. □

For p = 1, Lemma 2.7 reduces to Lemma 2.1 of [6]. For the reader’s convenience we recall here the definitions of the Gamma and Beta functions

Γ(x) = ∫_{0}^{∞} e^{−x} t^{x−1} dt, B(x, y) = ∫_{0}^{1} t^{x−1} (1 − t)^{y−1} dt.

It holds

B(x, y) = \frac{Γ(x)Γ(y)}{Γ(x + y)} .

The integral form of the hypergeometric function is

2F_{1}(x, y; c; z) = \frac{1}{B(y, c − y)} ∫_{0}^{1} t^{y−1} (1 − t)^{c−y−1} (1 − zt)^{−x} dt

for |z| < 1, c > y > 0.

254 NOOR, AWAN, MIHAI, AND NOOR

3. Main results

In this section, we derive our main results. Theorem 3.1. Let f : I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0}

with a < b and f′ ∈ L[a, b]. If |f′| is p-convex function, then

| \frac{f(a) + f(b)}{2} − \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx| ⩽ b^{1−p} · \frac{b^{p} − a^{p}}{2p} \{K_{1}|f′(a)| + K_{2}|f′(b)|\}.

where

(3.1)  K_{1} = \frac{2}{3} ·_{2}F_{1}( 1 − \frac{1}{p}, 3; 4; 1 − \frac{a^{p}}{b^{p}}) − \frac{1}{2} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 3; 1 − \frac{a^{p}}{b^{p}})

+ \frac{1}{12} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ,

(3.2) K_{2} = \frac{1}{3} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; 1 − \frac{a^{p}}{b^{p}}) − \frac{1}{2} ·_{2}F_{1}( 1 − \frac{1}{p}, 1; 3; 1 − \frac{a^{p}}{b^{p}})

+ \frac{1}{2} ·_{2}F_{1}( 1 − \frac{1}{p}, 1; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) − \frac{1}{12} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) .

Proof. Using Lemma 2.4 and the fact that |f′| is a p-convex function, we have

| \frac{f(a) + f(b)}{2} − \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx| (3.3)

= | \frac{b^{p} − a^{p}}{2p} ∫_{0}^{1} M_{p}^{−1} (a, b; t)(1 − 2t)f′(M_{p}(a, b; t)) dt|

⩽ \frac{b^{p} − a^{p}}{2p} ∫_{0}^{1} \frac{|1 − 2t|}{[ta^{p} + (1 − t)b^{p}]^{1− \frac{1}{p}}} |f′([ta^{p} + (1 − t)b^{p}]^{\frac{1}{p}} )|dt

⩽ \frac{b^{p} − a^{p}}{2p} ∫_{0}^{1} \frac{|1 − 2t|}{[ta^{p} + (1 − t)b^{p}]^{1− \frac{1}{p}}} [t|f′(a)| + (1 − t)|f′(b)|]dt

= \frac{b^{p} − a^{p}}{2p} (|f′(a)|I_{1} + |f′(b)|I_{2}),

where

(3.4) I_{1} = ∫_{0}^{1} \frac{|1 − 2t|t}{[ta^{p} + (1 − t)b^{p}]^{1−1/p}} dt = b^{1−p} [ \frac{2}{3} ·_{2}F_{1}( 1 − \frac{1}{p}, 3; 4; 1 − \frac{a^{p}}{b^{p}})

− \frac{1}{2} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 3; 1 − \frac{a^{p}}{b^{p}}) + \frac{1}{12} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ],

(3.5) I_{2} = ∫_{0}^{1} \frac{|1 − 2t|(1 − t)}{[ta^{p} + (1 − t)b^{p}]^{1−1/p}} dt

= b^{1−p} [ \frac{1}{3} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; 1 − \frac{a^{p}}{b^{p}}) − \frac{1}{2} ·_{2}F_{1}( 1 − \frac{1}{p}, 1; 3; 1 − \frac{a^{p}}{b^{p}})

+ \frac{1}{2} ·_{2}F_{1}( 1 − \frac{1}{p}, 1; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) − \frac{1}{22} ·_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ].

Introducing relations (3.4) and (3.5) in (3.3) completes the proof. □

Theorem 3.2. Let f: I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0} with a < b and f′ ∈ L[a, b]. If |f′|^{q} is p-convex function where q ⩾ 1, then

| \frac{f(a) + f(b)}{2} − \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx|

⩽ b^{1−p} · \frac{b^{p} − a^{p}}{2p} H^{1− \frac{1}{q}} \{K_{1}|f′(a)|^{q} + K_{2}|f′(b)|^{q}\}^{\frac{1}{q}} .

where K_{1}, K_{2} are given by (3.1) and (3.2) and

H = _{2}F_{1}( 1 − \frac{1}{p}, 2; 3; 1 − \frac{a^{p}}{b^{p}}) −_{2}F_{1}( 1 − \frac{1}{p}, 1; 2; 1 − \frac{a^{p}}{b^{p}})

+_{2}F_{1}( 1 − \frac{1}{p}, 1; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) .

Proof. Using Lemma 2.4, the fact that |f′| is a p-convex function and power mean’s inequality, we have

| \frac{f(a) + f(b)}{2} − \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx|

= | \frac{b^{p} − a^{p}}{2p} ∫_{0}^{1} M_{p}^{−1} (a, b; t)(1 − 2t)f′(M_{p}(a, b; t)) dt|

⩽ \frac{b^{p} − a^{p}}{2p} ( ∫_{0}^{1} \frac{|1 − 2t|}{[ta^{p} + (1 − t)b^{p}]^{1− \frac{1}{p}}} dt)^{1− \frac{1}{q}}

× ( ∫_{0}^{1} \frac{|1 − 2t|}{[ta^{p} + (1 − t)b^{p}]^{1− \frac{1}{p}}} |f′([ta^{p} + (1 − t)b^{p}]^{\frac{1}{p}} )|^{q}dt) ^{\frac{1}{q}}

⩽ \frac{b^{p} − a^{p}}{2p} ( ∫_{0}^{1} \frac{|1 − 2t|}{[ta^{p} + (1 − t)b^{p}]^{1− \frac{1}{p}}} dt)^{1− \frac{1}{q}}

× ( ∫_{0}^{1} \frac{|1 − 2t|}{[ta^{p} + (1 − t)b^{p}]^{1− \frac{1}{p}}} [t|f′(a)|^{q} + (1 − t)|f′(b)|^{q}]dt) ^{\frac{1}{q}}

= b^{1−p} · \frac{b^{p} − a^{p}}{2p} H^{1− \frac{1}{q}} \{K_{1}|f′(a)|^{q} + K_{2}∥f′(b)|^{q}\}^{\frac{1}{q}} .  □

Theorem 3.3. Let f: I = [a, b] ⊂ ℝ → ℝ be a differentiable function on I^{0} with a < b and f′ ∈ L[a, b]. If |f′| is p-convex function, then

| \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx − f( [ \frac{a^{p} + b^{p}}{2} ]^{\frac{1}{p}} ) |

⩽ \frac{b^{1−p} (b^{p} − a^{p})}{p}[\{C_{1} + C_{2} − C_{3}\}|f′(a)| + \{C_{4} + C_{5} − C_{6} − C_{7}\}|f′(b)|],

where

C_{1} = \frac{1}{6}_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; 1 − \frac{a^{p}}{b^{p}}) ,

256 NOOR, AWAN, MIHAI, AND NOOR

C_{2} = \frac{1}{12}_{2}F_{1}( 1 − \frac{1}{p}, 3; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ,
C_{3} = \frac{1}{48}_{2}F_{1}( 1 − \frac{1}{p}, 2; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ,
C_{4} = \frac{1}{3}_{2}F_{1}( 1 − \frac{1}{p}, 1; 4; 1 − \frac{a^{p}}{b^{p}}) ,
C_{5} = \frac{3}{8}_{2}F_{1}( 1 − \frac{1}{p}, 2; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ,
C_{6} = \frac{1}{12}_{2}F_{1}( 1 − \frac{1}{p}, 3; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) ,
C_{7} = \frac{1}{2}_{2}F_{1}( 1 − \frac{1}{p}, 1; 2; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) .

Proof. Using Lemma 2.7 and the fact that |f′| is a p-convex function, we have

| \frac{p}{b^{p} − a^{p}} ∫_{a}^{b} \frac{f(x)}{x^{1−p}} dx − f( [ \frac{a^{p} + b^{p}}{2} ]^{\frac{1}{p}} ) |

= | \frac{b^{p} − a^{p}}{p} ∫0^{\frac{1}{2}} tM_{p}^{−1} (a, b; t)f′(M_{p}(a, b; t)) dt

+ \frac{b^{p} − a^{p}}{p} ∫ 1_{\frac{1}{2}}(t − 1)M_{p}^{−1} (a, b; t)f′(M_{p}(a, b; t)) dt|

⩽ \frac{b^{p} − a^{p}}{p} [ ∫0^{\frac{1}{2}} tM_{p}^{−1} (a, b; t)|f′(M_{p}(a, b; t))|dt

+ ∫ 1_{\frac{1}{2}} |t − 1|M_{p}^{−1} (a, b; t)|f′(M_{p}(a, b; t))|dt]

⩽ \frac{b^{1−p} (b^{p} − a^{p})}{p}[\{ \frac{1}{6}_{2}F_{1}( 1 − \frac{1}{p}, 2; 4; 1 − \frac{a^{p}}{b^{p}})

+ \frac{1}{12}_{2}F_{1}( 1 − \frac{1}{p}, 3; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) )

− \frac{1}{48}_{2}F_{1}( 1 − \frac{1}{p}, 2; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) \}|f′(a)|

+ \{ \frac{1}{3}_{2}F_{1}( 1 − \frac{1}{p}, 1; 4; 1 − \frac{a^{p}}{b^{p}}) + \frac{3}{8}_{2}F_{1}( 1 − \frac{1}{p}, 2; 3; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) )

− \frac{1}{12}_{2}F_{1}( 1 − \frac{1}{p}, 3; 4; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) )

− \frac{1}{2}_{2}F_{1}( 1 − \frac{1}{p}, 1; 2; \frac{1}{2}( 1 − \frac{a^{p}}{b^{p}}) ) \}|f′(b)|].

This completes the proof. □

Acknowledgement. The authors would like to thank anonymous referee for his/her valuable comments and suggestions.

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Department of Mathematics (Received 07 12 2014) COMSATS Institute of Information Technology Islamabad, Pakistan

noormaslam@gmail.com
khalidanoor@hotmail.com

Department of Mathematics Government College University Faisalabad, Pakistan

awan.uzair@gmail.com

Department scientific-methodical sessions Romanian Mathematical Society-branch Bucharest Bucharest, Romania

marcelamihai58@yahoo.com