Several integral inequalities for (α, s,m)-convex functions

1.   Introduction

The general theory of convex functions is the origin of powerful tools for the study of problems in analysis. Convex functions are thoroughly associated to the theory of inequalities. Inequalities involving convex functions are the most efficient tools in the development of several branches of mathematics and has been given considerable attention in the literature (see ^[1,2]) and references therein). We start by giving some renowned definitions:

Definition 1.1. A function $f:I\subseteq R \rightarrow R\;$ is said convex if

for all $a, b\in I\;$and $t\in \left[ 0, 1\right].$

In ^[3] Toader defines the $m-$convexity:

Definition 1.2. A function$\; f:[0, d]\rightarrow R_{0} = [0, \infty)$ is said $m-$convex on $[0, d]$ for some $m\in (0, 1], \;$if

In ^[4] Miheşan introduced the following class of functions:

Definition 1.3. A function$\; f:[0, d]\rightarrow R_{0} = [0, \infty)$ is said $\left(\alpha, m\right)-$ convex on $[0, d], \; (\alpha, m)\in (0, 1]^{2}$, if

Let $K_{\alpha }^{m}$ be the set of $\left(\alpha, m\right)-\;$convex functions on $\left[ 0, d\right].$

In ^[5] Park J. introduced the following class of $(s, m)-$convex functions in the second sense:

Definition 1.4. For some fixed $s\in \left(0, 1\right]$ and $m\in \left[ 0, 1\right]$ a mapping $f:[0, d] \rightarrow \mathbb{R}$ is said $(s, m)-$ convex in the second sense on $[0, d]$, if

holds for all $a, b\in [0, d]$ and $t\in \left[ 0, 1\right]$.

In ^[6], Özdemir et al. gave the following lemmas for twice differentiable functions.

Lemma 1.1. Let $f:I\subseteq R\rightarrow R$ be a twice differentiable mapping on $I^{\circ}$(interior of $I$), $a\neq b\in I$ and $f^{\prime \prime} \in L[a, b]$, then the following equality holds:

Lemma 1.2. Let $f:I\subseteq R\rightarrow R$ be a twice differentiable mapping on $I^{\circ}$, where $a, b\in I$ with $a < mb$ and $m\in (0, 1].\;$If $f^{\prime \prime } \; \in L[a, b], \;$then the following equality holds:

In ^[7], M. Z. Sarıkaya and N. Aktan obtained the following Trapezoid inequality for convex functions

In ^[8], I. Işcan gave a refinement of the Hölder's integral inequality as follows:

Theorem 1.1. (Hölder- İşcan integral inequality) Let $p > 1$ and $q^{-1}+p^{-1} = 1.$ If $f$ and $g$ are real functions defined on interval $[a, b]$ and if $|f|^p$ and $|g|^q$ are integrable functions on $[a, b]$, then

An refinement of power-mean integral inequality as a different version of the Hölder- İşcan integral inequality can be given as follows:

Theorem 1.2. (Improved power-mean integral inequality ^[9]) Let $q\ge1$. If $f$ and $g$ are real functions defined on $[a, b]$ and if $\left\vert f\right\vert, \; \left\vert f\right\vert \left\vert g\right\vert ^{q}$ are integrable functions on $[a, b]$, then

There is a massive literature on Hermite – Hadamard type inequalities via different kind of convexities. The most notable are $m$–convex, $\left(\alpha, m\right)-$convex, $s$ –convex and extended $s-$ convex (see ^{[10,11,12,13,14,15,16,17,18,19]}). Besides this, we mention recent results related to the Hermite–Hadamard type inequalities, for example, see ^[20,21,22] and the references therein. In the next section we consider the most recent generalized convex functions and its properties which covers all the above as particular cases.

2.   Some properties of $\left(\alpha, s, m\right)$–convex functions

Definition 2.1. ^[23] The function $f:\left[ 0, d\right] \rightarrow R$ is said to be $\left(\alpha, s, m\right) -$convex, if we have

where $a, b\in \left[ 0, d\right] \;$, $t\in \left(0, 1\right) \;$and for some $s\in \left[ -1, 1\right], \; \left(\alpha, m\right) \in \left(0, 1\right] ^{2}.$

Here $K_{\alpha }^{s, m}$ is the set of $\left(\alpha, s, m\right) \;$ convex functions on $\left[ 0, d\right].$

Remark 2.1. (i) If $s = 1$, then $f$ is an $(\alpha, m)$ - convex function on $(0, d]$.

(ii) If $\alpha = 1,$ then $f$ is an extended $(s, m)$ - convex function on $(0, d]$.

(iii) If $\alpha = m = 1$, then $f$ is an extended s -convex function on $(0, d]$.

(iv) If $\ \alpha = s = m = 1$, then f is a convex function on $(0, d].$

Proposition 2.1. If a function $f\;$ is $\left(\alpha, m\right)$ – convex for all $s\in[-1, 1]$, then it is also $\left(\alpha, s, m\right)$ – convex.

Proof. Since $f\in K_{\alpha }^{m}$, we have

$\forall \; a, b\in \left[ 0, d\right], \; t\in \left[ 0, 1\right] \;$and $\left(\alpha, m\right) \in \left(0, 1\right] ^{2}.\;$ On the other hand, we have $t^{\alpha }\leq t^{\alpha s}\;$and $1-t^{\alpha }\leq \left(1-t^{\alpha }\right) ^{s}, \;$for $s\in \left[ -1, 1\right], \;$thus

$i.e\; f\in K_{\alpha }^{s, m}.$

Proposition 2.2. Let $\alpha \in \left(0, 1\right] \;$and $s\in \left[ -1, 1 \right], \;$then

$\left(i\right) \;$when $s\in \left(-1, 1\right], \;$we have

$\left(ii\right) \;$when$\; s = -1\;$, we have $\int_{0}^{1}\frac{t\left(1-t\right) }{\left(1-t^{\alpha }\right) }dt = \frac{ 1}{\alpha }\left\{ \Psi \left(\frac{3}{\alpha }\right) -\Psi \left(\frac{2 }{\alpha }\right) \right\},$

where $\Gamma \left(.\right), \; \beta \left(., .\right), \; \Psi \left(.\right) \;$are the classical Euler-Gamma, Beta and Psi-Gamma functions, respectively.

Proof. $\left(i\right)$

If we change the variable $t^{\alpha } = z\Rightarrow t = z^{\frac{1}{\alpha } }\; ,$ we get $dt = \frac{1}{\alpha }z^{\frac{1}{\alpha }-1}dz.\;$ Hence

which completes the proof of proposition $\left(i\right).$

$\left(ii\right) \;$When$\; s = -1\;$

Again using the change of variable $t^{\alpha } = z\Rightarrow t = z^{\frac{1}{ \alpha }}\;$, we have

where $\Psi \left(x\right) +\gamma = \int_{0}^{1}\frac{1-t^{x-1}}{1-t}dt\;$ and $\gamma = \int_{0}^{\infty }\left(\frac{1}{1+t}-e^{-t}\right) \frac{1}{t} \ dt\;$is Euler–Mascheroini constant (see p.258 ^[24]). Which completes the proof of proposition $\left(ii\right).$

3.   Hermite-Hadamard type integral inequalities for $(\alpha, s, m)$-convex functions

Theorem 3.1. If $f:\left[ 0, d\right] \rightarrow R \;$is $\left(\alpha, s, m\right) -$convex, we have

where $a, b\in \left[ 0, d\right] \;$, $a < mb \; and\, \, for \, \, some \; s\in \left[ 0, 1 \right], \; \left(\alpha, m\right) \in \left(0, 1\right] ^{2}.$

Proof. The left-hand side of (3.1) is easy to prove:

The proof of the right-hand side (3.1).

By the $\left(\alpha, s, m\right)-$convexity of $f\;$we observe that

Integrating the resulting inequality with respect to $t$, we get

as

and

which proved the theorem as we used the fact

Remark 3.1. If we take $\alpha = m = s = 1.\;$The function $f\;$becomes $\left(1, 1, 1\right)-$convex. Hence the inequality $\left(3.1\right)$ reduces to the Hermite- Hadamard inequality :

Theorem 3.2. Let $\ f:I\subset (0, d]\rightarrow R$ be a differentiable function on $I^{\circ}(I^{\circ}\;$is interior of $I\; )$ for $a, b\in I^{\circ}$ such that $\; f^{\prime \prime }\in L[a, b]$ with $0 < a < mb$. If $\left\vert f^{\prime \prime }\right\vert$ is an $\left(\alpha, s, m\right)-$convex function on $[a, b]$, then the following inequality holds:

where$\; (\alpha, m)\in (0, 1]^{2}, s\in (0, 1].$

Proof. From Lemma 1.2 and using the $\left(\alpha, s, m\right) -$convexity of $\left\vert f^{\prime \prime }\right\vert,$ we have

where we used the facts that, since $t^{\alpha }\geq t\;$for all $\alpha \in \left(0, 1\right] \;$and $t\in \left[ 0, 1\right], \;$we have $-t^{\alpha }\leq -t\;$or$\; 1-t^{\alpha }\leq 1-t\;$, $1-t^{\alpha }\leq \left(1-t\right) ^{\alpha }$and $\left(1-t^{\alpha }\right) ^{s}\leq \left(1-t\right) ^{\alpha s}, s\in \left(0, 1\right].\;$If we multiply the resulting inequality with $t\left(1-t\right) \;$and taking integral on $\left[ 0, 1\right], \;$we have

and

Corollary 3.1. From Remark 2.1 (iv), since f(.) is a convex function on $[a, b]$ for $m = \alpha = s = 1,$ we obtain the inequality (1.2).

Corollary 3.2. Under the same conditions in Theorem 3.2 with $\left\vert f^{\prime \prime }\left(x\right) \right\vert \leq M$ for all $x\in[a, b]$, we have

The following Theorem presents a upper bound for $\left(\alpha, s, m\right)$-convex functions.

Theorem 3.3. Let $\ f:I\subset (0, d]\rightarrow R$ be a differentiable function on $I^{\circ}(I^{\circ}\;$is interior of $I\; )$ for $a, b\in I^{\circ}$, $\; f^{\prime \prime }\in L[a, b], \; (\alpha, m)\in (0, 1]^{2}\; ,s\in (-1, 1]\;$ with $0 < a < mb$. If $\left\vert f^{\prime \prime }\right\vert ^{q}$ is an $\left(\alpha, s, m\right) -$convex function on $[a, b]$ for $q\geq 1,$ then the following inequality holds:

where $M_{\beta }\left(\alpha, s\right)$ is given in Proposition 2.2(i).

Proof. First, we assume that $q = 1.\;$From Lemma 1.1 and using the$\;$ $\left(\alpha, s, m\right)-$convexity with properties of modulus we have

which completes the proof for q = 1. Here we also used the Proposition 2.2 (ⅰ).

Secondly, suppose now that $q > 1$. From Lemma 1 and using the Hölder's integral inequality for $q > 1$ with properties of modulus, we have

where $p^{-1}+q^{-1} = 1.$

Since $\ \left\vert f^{\prime \prime }\right\vert ^{q}\;$is $(\alpha, s, m)-$convex on $[a, b]$, we know that for every $t\in \lbrack 0, 1].$

From (3.4), (3.5) and Proposition 2.2 (ⅰ)

which completes the inequality (3.3).

Corollary 3.3. Under the same conditions in Theorem 3.3 with $\left\vert f^{\prime \prime }\left(x\right) \right\vert \leq M$ and $m = 1$ for all $x\in[a, b]$, we have

Theorem 3.4. When $s = -1, \ $under the same conditions of Theorem 3.3, we obtain the following result

Proof. Using the $\left(\alpha, s, m\right)-$convexity of $\left\vert f^{\prime \prime }\right\vert^q \;$ and Proposition 2.2(ⅱ) with the properties of modulus, we have

which completes the inequality (3.6).

Corollary 3.4. Under the same conditions in Theorem 3.4 with $\left\vert f^{\prime \prime }\left(x\right) \right\vert \leq M$ and $m = 1$ for all $x\in \lbrack a, b]$, we have

Theorem 3.5. Let $\ f:I\subset (0, d]\rightarrow R$ be a twice differentiable function on $I^{\circ}$ for $a, b\in I^{\circ}$ such that $\; f^{\prime \prime }\in L[a, b], \; (\alpha, m)\in (0, 1]^{2}, s\in (-1, 1]\;$with $0 < a < mb$. If $\left\vert f^{\prime \prime }\right\vert ^{q}$ is an $\left(\alpha, s, m\right) -$convex function on $[a, b]\;$for $q\ge1\ and\; q\geq r > 0,$ then the following inequality holds:

Proof. Using the Lemma 1.1, Hölder's inequality and $\left(\alpha, s, m\right) -$convexity of $\left\vert f^{\prime \prime }\right\vert^q \;$ we have

After simplifying we get inequality (3.7).

An immediate consequences of Theorem 3.5 by considering special cases for $(r = 0)$ and $(m = 1)$ can be given as:

Corollary 3.5. Under the assumptions of Theorem 3.5, the following inequality holds

Corollary 3.6. Under the assumptions of Theorem 3.5, the following inequality holds

Corollary 3.7. Under the same conditions in Theorem $3.5$ with $\left\vert f^{\prime \prime }\left(x\right) \right\vert\leq M$ and $m = 1$, for all $x\in \lbrack a, b]$, we have

Theorem 3.6. Let $\ f:I\subset (0, d]\rightarrow R$ be a twice differentiable function on $I^{\circ}$ such that $\; f^{\prime \prime }\in L[a, b]$ for $a, b\in I^{\circ}, \; (\alpha, m)\in (0, 1]^{2}, s\in (-1, 1]\;$ with $0 < a < mb$. If $\left\vert f^{\prime \prime }\right\vert ^{q}\;$is an $\left(\alpha, s, m\right)-$convex function on $[a, b]\;$for $q\ge1, \; q\geq r > 0$, then the following inequality holds:

where $\ K = \frac{\left(b-a\right) ^{2}}{2}.$

Proof. By using Lemma 1.1 and Hölder's inequality with $\left(\alpha, s, m\right)-$convexity of $\left\vert f^{\prime \prime }\right\vert ^{q}$, we have

Here we used the fact that

and with a formal change of variables $t^{\alpha } = z$

If we write the above integrals places we obtain the inequality (3.10).

Theorem 3.7. Let $\ f:I\subset (0, d]\rightarrow R$ be a twice differentiable function on $I^{\circ}$ such that $\; f^{\prime \prime }\in L[a, b]$ for $a, b\in I^{\circ}, \; (\alpha, m)\in (0, 1]^{2}, s\in [0, 1]\;$ with $0 < a < mb$. Also let $p > 1$ such that $q = \frac{p}{p-1}$, if $\left\vert f^{\prime \prime }\right\vert ^{q}$ is an $\left(\alpha, s, m\right) -$ convex on $[a, b]$, then the following inequality holds:

where $G = \frac{\left(mb-a\right) ^{2}}{2}.$

Proof. Using Lemma 1.2 and Hölder- İşcan integral inequality along with $\left(\alpha, s, m\right) -$convexity of $\left\vert f^{\prime \prime }\right\vert ^{q}$, we have

Using the fact that $\left\vert m-n\right\vert ^{c}\leq m^{c}-n^{c}$ for $c > 1$ and for $m, n > \ 0; \ \ m > n$. The right-hand side of above inequality becomes

Simplifying above integrals, we obtain the result of the Theorem 3.7.

Corollary 3.8. Under the same conditions in Theorem 3.7 with $\left\vert f^{\prime \prime }\left(x\right) \right\vert\leq M$ for all $x\in \lbrack a, b]$, we have

Theorem 3.8. Let $\ f:I\subset (0, d]\rightarrow R$ be a twice differentiable function on $I^{\circ}$ such that $\; f^{\prime \prime }\in L[a, b]$ for $a, b\in I^{\circ}, \; (\alpha, m)\in (0, 1]^{2}, s\in [0, 1]\;$ with $0 < a < mb$. If $\left\vert f^{\prime \prime }\right\vert ^{q}$ is an $\left(\alpha, s, m\right) -$ convex on $[a, b]$ for $q\ge1$, then the following inequality holds:

where $G = \frac{\left(mb-a\right) ^{2}}{2}.$

Proof. Using Lemma 1.2 and improved power mean integral inequality with $\left(\alpha, s, m\right)-$convexity of $\left\vert f^{\prime \prime }\right\vert ^{q}$, we have

By employing $\left(\alpha, s, m\right)-$convexity and simplifying, we have

It can be noticed that

and

and

Replacing the values of the integrals computed in (3.14), (3.15), (3.16) and (3.17) in (3.13), we get (3.12).

Corollary 3.9. Under the same conditions in Theorem $3.8$ with $\left\vert f^{^{\prime \prime }}\left(x\right) \right\vert\leq M$ for all $x\in \lbrack a, b]$, we have

4.   Application to special means

Now we let us consider some special means for arbitrary positive real numbers $c$ and $d$.

$The\ arithmetic\ mean\ \ \ :$

$\ \ \ \ \ \ \ A\left(c, d\right) = \frac{c+d}{2}, \qquad \qquad c, d\in R\ with\ c, d > 0.$

$The\ geometric\ mean\ \ \ :$

$G\left(c, d\right) = \left(c \cdot d\right) ^{\frac{1}{2}}, \qquad c, d\in R\ with\ c, d > 0.$

$The\ harmonic\ mean\ \ \ :$

$H\left(c, d\right) = \frac{2\ c\ d}{c + d}, \qquad \qquad c, d\in R\ \backslash \{0\}.$

$The\ logarithmic\ mean\ \ \ :$

$L\left(c, d\right) = \left\{ \begin{array}{c} c, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ c = d \\ \frac {d-c}{ln d - ln c}, \ \ \ \ \ \ \ \ if\ \ c\neq d. \end{array} \ \ \ \ \right.$

$The\ Generalized\ logarithmric\ mean\ \ \ :$

$L_{n}\left(c, d\right) = \left\{ \begin{array}{c} c, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ c = d, \\ \left(\frac{d^{n+1}-c^{n+1}}{\left(n+1\right) \left(d-c\right) }\right)^{1/n}, \ \ \ \ \ \ \ \ if\ \ c\neq d, \ \ n\in Z\backslash \{-1, 0\}; c, d > 0. \end{array} \ \ \ \ \right.$

$The\ \ Identric\ mean\ \ \ :$

$I\left(c, d\right) = \left\{ \begin{array}{c} c, \qquad if\ \ c = d \\ \frac{1}{e}\left(\frac{d^{d}}{c^{c}}\right) ^{\frac{1}{d-c}}, \qquad if\ \ c\neq d \; \ c, d > 0. \end{array} \ \ \ \ \right.$

Proposition 4.1. Let $a, b\in R, \ \ a < b\ $, $\ a, b > 0\ $, then we have the following inequality holds

Proof. This assertion follows from Theorem 3.3 for $f\left(x\right) = e^{x}$ and $\ x\in R$.

Proposition 4.2. Let $a, b\in R, \ \ a < b$, $a, b > 0$, then we have the following inequality

Proof. This assertion follows from Theorem 3.4 for $f\left(x\right) = -ln{x}$ and $\ x > 0$.

Proposition 4.3. Let $a, b\in R, \ \ a < b\ $ and $0\notin \left[ a, b\right]$, then the following inequality holds

Proof. This assertion follows from Theorem 3.4 for $f\left(x\right) = \frac{1}{x}$; $\ x \in[a, b]$.

Proposition 4.4. Let $a, b\in R, \ \ a < b$ and $0\notin \left[ a, b\right]$, then the following inequality holds

Proof. This assertion follows from Theorem 3.5 for $f\left(x\right) = \frac{1}{x}$; $x \in[a, b]$.

5.   Conclusions

In many practical studies, it is necessary to evaluate the difference between the two quantities. From the point of view of programming an algorithm for solving optimization problems, classical inequalities with the smallest upper limit play an important role.

We have proved several new generalized integral inequalities involving $\left(\alpha, s, m\right)-$convex functions for extended case of s where $s \in[-1, 1]$, we obtained Hermite–Hadamrad type inequalities. The most interesting case is extended case for $s = -1$, when we get connected to Psi-Gamma functions. We analyze new upper bounds involving special functions by employing different variants of Hölder inequality such as Hölder-Işcan inequality and Improved Power mean inequality.

Conflict of interest

The author declares no conflict of interest in this paper.