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Cauchy type means for some generalized convex functions

Abstract

In this paper, we establish Jensen’s inequality for s-convex functions in the first sense. By using Jensen’s inequalities, we obtain some Cauchy type means for p-convex and s-convex functions in the first sense. Also, by using Hermite–Hadamard inequalities for the respective generalized convex functions, we find new generalized Cauchy type means.

Introduction

Cauchy mean value theorem is of huge importance in mathematical analysis. Mercer [18] and Pečarić [21] made connection between Cauchy type means and Jensen’s inequality. These are given both in discrete and in integral form and have many applications. A meaningful advancement in theory of Cauchy type means is given in [15, 1821]. Also see [811, 1517] for more information about means. The following result is given in [19], which involves Jensen’s inequality both in numerator and denominator.

Theorem 1.1

([19])

Let \(G\subseteq \mathbb{R}\) be an interval and \(r_{i}>0\) for all \(1\leq i\leq n\) such that \(\Sigma ^{n}_{i=1}r_{i}=S_{n}\) and \(c_{1},\ldots,c_{n}\in G\) not all the same. Consider the twice differentiable functions \(\zeta _{1},\zeta _{2}:G\rightarrow \mathbb{R}\) such that

$$0\leq l\leq \zeta _{1}^{\prime \prime }(c)\leq L\quad \textit{and} \quad 0\leq m\leq \zeta _{2}^{\prime \prime }(x) \leq M \quad \textit{for all } c\in G. $$

Then

$$ \frac{l}{M}\leq \frac{\frac{1}{S_{n}}\Sigma ^{n}_{i=1}r_{i}\zeta _{1}(c_{i})-\zeta _{1}(\frac{1}{S_{n}}\Sigma ^{n}_{i=1}r_{i}c_{i})}{\frac{1}{S_{n}}\Sigma ^{n}_{i=1}r_{i}\zeta _{2}(c_{i})-\zeta _{2}(\frac{1}{S_{n}}\Sigma ^{n}_{i=1}r_{i}c_{i})} \leq \frac{L}{m}. $$
(1)

Here our aim is to find some Cauchy type means for p-convex and s-convex functions in the first sense using Jensen’s and Hermite–Hadamard inequalities, respectively.

Let M, N be two bivariable means defined in a real interval G, and let \(J\subseteq G \) be a subinterval of G. According to Aumann [6], a function \(\zeta :J\rightarrow G\) is convex with respect to the pair of means \((M,N)\) if

$$ \zeta \bigl(M(j_{1},j_{2})\bigr)\leq N\bigl(\zeta (j_{1}),\zeta (j_{2})\bigr), \quad j_{1},j_{2} \in J; $$

and ζ is convex with respect to M if

$$ \zeta \bigl(M(j_{1},j_{2})\bigr)\leq M\bigl(\zeta (j_{1}),\zeta (j_{2})\bigr), \quad j_{1},j_{2} \in J. $$

These notions generalize the classical notions of convexity. Moreover, taking for M the weighted power mean, i.e.,

$$ M(j_{1},j_{2})=\bigl[rj_{1}^{p}+(1-r)j_{2}^{p} \bigr]^{\frac{1}{p}}, $$

and for N the weighted arithmetic mean

$$ N(j_{1},j_{2})=[rj_{1}+(1-r)j_{2}, $$

one gets the following definition.

Definition 1.1

([13, 14])

Let \(G\subset (0, \infty )\) be a real interval and \(p\in \mathbb{R\backslash }\{0\}\). A function \(\zeta :G\rightarrow \mathbb{R}\) is said to be a p-convex function if

$$ \zeta \bigl[ \bigl[rg_{1}^{p}+(1-r)g_{2}^{p} \bigr]^{\frac{1}{p}} \bigr] \leq r \zeta (g_{1})+(1-r)\zeta (g_{2}) $$
(2)

for all \(g_{1},g_{2}\in G\) and \(r\in [0,1]\). If inequality (2) is reversed, then ζ is called p-concave function.

Definition 1.2

([12])

Let \(s\in (0,1]\). A function \(\zeta :[0,\infty )\rightarrow \mathbb{R}\) is called an s-convex function (in the first sense) or \(\zeta \in K^{1}_{s}\) if

$$ \zeta (r_{1} g_{1}+r_{2} g_{2})\leq r_{1}^{s}\zeta (g_{1})+r_{2}^{s} \zeta (g_{2}) $$
(3)

for all \(g_{1},g_{2}\in \mathbb{R}^{+}=[0,\infty )\) and \(r_{1},r_{2}\geq 0\) with \(r_{1}^{s}+r_{2}^{s}=1\).

Cauchy type means for p-convex functions in Jensen’s sense

Toplu et al. [22] proved Jensen’s inequality for p-convex functions as follows.

Theorem 2.1

([22])

Let \(p\in \mathbb{R\backslash }\{0\}\) and \(\zeta :G\subset (0,\infty )\rightarrow \mathbb{R}\) be a p-convex function. Let \(g_{i}\in G\) and \(r_{i}\in [0,1]\), \(0\leq i\leq n \), then the following inequality holds:

$$ \zeta \Biggl( \Biggl( \sum^{n}_{1}r_{i}g_{i}^{p} \Biggr) ^{ \frac{1}{p}} \Biggr) \leq \sum^{n}_{1}r_{i} \zeta (g_{i}), $$
(4)

where \(\sum^{n}_{1}r_{i}=1\).

Now, by using Theorem 2.1, we state and prove the following theorem, which gives the Cauchy type mean for p-convex function.

Theorem 2.2

Let \(G\subset (0,\infty )\) be an interval, \(p\in \mathbb{R\backslash }\{0\}\), and \(r_{i}\in [0,1]\). Let \(\zeta _{1},\zeta _{2}\in C^{2}(G)\) be p-convex functions. Then there exist some \(\chi \in G\) such that the following equality holds:

$$ \frac{\sum^{n}_{1}r_{i}\zeta _{1}(g_{i})-\zeta _{1} ( ( \sum^{n}_{1}r_{i}g^{p}_{i} ) ^{\frac{1}{p}} ) }{\sum^{n}_{1}r_{i}\zeta _{2}(g_{i})-\zeta _{2} ( (\sum^{n}_{1}r_{i}g^{p}_{i})^{\frac{1}{p}} ) } =\frac{\zeta ^{\prime \prime }_{1}(\chi )}{\zeta ^{\prime \prime }_{2}(\chi )}, $$
(5)

with each \(r_{i}\in [0,1]\) such that \(\sum^{n}_{1}r_{i}=1\) and provided that the denominators are non-zero.

Proof

Let us define

$$ H:= \Biggl( \sum^{n}_{1}r_{i}g^{p}_{i} \Biggr) ^{\frac{1}{p}} $$

and

$$ (T\zeta _{1}) (\lambda ):=\sum^{n}_{1}r_{i} \zeta _{1}\bigl(\lambda g_{i}+(1- \lambda )H\bigr)-\zeta _{1}(H), $$

where \(\lambda \in [0,1]\). Similarly, we define \((T\zeta _{2})(\lambda )\).

Note that

$$ (T\zeta _{1})^{\prime }(\lambda ):=\sum ^{n}_{1}r_{i}(g_{i}-H) \zeta ^{\prime }_{1}\bigl( \lambda g_{i}+(1- \lambda )H\bigr) $$

and

$$ (T\zeta _{1})^{\prime \prime }(\lambda ):=\sum ^{n}_{1}r_{i}(g_{i}-H)^{2} \zeta ^{\prime \prime }_{1}\bigl( \lambda g_{i}+(1-\lambda )H\bigr). $$

Now consider a function \(Q(\lambda )\) defined as follows:

$$ Q(\lambda )=(T\zeta _{2}) (1) (T\zeta _{1}) (\lambda )-(T\zeta _{1}) (1) (T \zeta _{2}) (\lambda ), $$

such that we have

$$ Q(0)=Q(1)=Q^{\prime }(0)=0. $$

Then from two applications of mean value theorem, we have \(\upsilon \in G\) so that

$$ Q^{\prime \prime }(\upsilon )=0. $$

It implies that

$$ \sum_{i=1}^{n}r_{i}(g_{i}-H)^{2} \bigl[(T\zeta _{2}) (1)\zeta ^{\prime \prime }_{1}\bigl( \upsilon g_{i} +(1-\upsilon )H\bigr) -(T\zeta _{1}) (1) \zeta ^{\prime \prime }_{2}\bigl( \upsilon g_{i} +(1- \upsilon )H\bigr)\bigr]=0. $$
(6)

For some fixed υ, the expression in the square brackets in (6) is a continuous function of \(g_{i}\), so it vanishes. Corresponding to that value of \(g_{i}\), we can have a number

$$ \chi =\upsilon g_{i} +(1-\upsilon )H $$

such that

$$ (T\zeta _{2}) (1).\zeta ^{\prime \prime }_{1}(\chi )-(T \zeta _{1}) (1).\zeta ^{\prime \prime }_{2}( \chi )=0. $$

This gives equality (5). □

Corollary 2.3

Let \(G\subset (0,\infty )\) be an interval, \(p\in \mathbb{R\backslash }\{0\}\), and \(r_{i}\in [0,1]\). Let \(\zeta _{1},\zeta _{2}\in C^{2}(G)\) be p-convex functions such that \(\frac{\zeta ^{\prime \prime }_{1}}{\zeta ^{\prime \prime }_{2}}\) is invertible. Then there exist some \(\chi \in G\) such that the following equality holds:

$$ \chi = \biggl( \frac{\zeta ^{\prime \prime }_{1}}{\zeta ^{\prime \prime }_{2}} \biggr) ^{-1} \biggl( \frac{\sum^{n}_{1}r_{i}\zeta _{1}(g_{i})-\zeta _{1} ( (\sum^{n}_{1}r_{i}g^{p}_{i})^{\frac{1}{p}} ) }{\sum^{n}_{1}r_{i}\zeta _{2}(g_{i})-\zeta _{2} ( (\sum^{n}_{1}r_{i}g^{p}_{i})^{\frac{1}{p}} ) } \biggr) , $$
(7)

with each \(r_{i}\in [0,1]\) such that \(\sum^{n}_{1}r_{i}=1\) and provided that the denominators are non-zero.

Corollary 2.4

Let \(G\subset (0,\infty )\) be an interval, \(p\in \mathbb{R\backslash }\{0\}\), and \(r_{i}\in [0,1]\). Let \(\zeta \in C^{2}(G)\) be a p-convex function. Then there exist some \(\chi \in G\) such that the following equality holds:

$$ \sum^{n}_{1}r_{i} \zeta (g_{i})-\zeta \Biggl( \Biggl( \sum ^{n}_{1}r_{i}g^{p}_{i} \Biggr) ^{\frac{1}{p}} \Biggr) =\frac{\zeta ^{\prime \prime }(\chi )}{2} \Biggl( \sum ^{n}_{1}r_{i}g_{i}^{2}- \Biggl( \Biggl( \sum^{n}_{1}r_{i}g^{p}_{i} \Biggr) ^{\frac{1}{p}} \Biggr) ^{2} \Biggr) $$
(8)

with each \(r_{i}\in [0,1]\) such that \(\Sigma ^{n}_{1}r_{i}=1\).

Proof

By letting \(\zeta _{1}=\zeta \) and \(\zeta _{2}(w)=w^{2}\), where \(w\in (0,\infty )\), in Theorem 2.2, we achieve equality (8). □

Cauchy type means for p-convex functions in the Hermite–Hadamard sense

Let \(\zeta :G\subset (0,\infty )\rightarrow \mathbb{R}\) be a p-convex function, \(p\in \mathbb{R}\backslash \{0\}\), and \(g_{1},g_{2}\in G\) with \(g_{1}< g_{2}\). If \(\zeta \in L_{1}[g_{1},g_{2}]\), then we have (e.g., see [13])

$$ \zeta \biggl( \biggl( \frac{g_{1}^{p}+g_{2}^{p}}{2} \biggr) ^{ \frac{1}{p}} \biggr) \leq \frac{p}{g_{2}^{p}-g_{1}^{p}} \int ^{g_{2}}_{g_{1}} \frac{\zeta (w)}{w^{1-p}}\,dw\leq \frac{\zeta (g_{1})+\zeta (g_{2})}{2}. $$
(9)

By using the right half of inequality (9), we have following result.

Theorem 3.1

Let \(G\subset (0,\infty )\) be an interval, \(p\in \mathbb{R}\backslash \{0\}\), and \(g_{1},g_{2}\in G\) with \(g_{1}< g_{2}\). Let \(\zeta _{1},\zeta _{2}\in C^{2}(G)\) be p-convex functions. Then there exists some \(\chi \in G\) such that the following equality holds:

$$ \frac{\frac{p}{g_{2}^{p}-g_{1}^{p}}\int ^{g_{2}}_{g_{1}}\frac{\zeta _{1}(w)}{w^{1-p}}\,dw-\zeta _{1} ( ( \frac{g_{1}^{p}+g_{2}^{p}}{2} ) ^{\frac{1}{p}} ) }{\frac{p}{g_{2}^{p}-g_{1}^{p}}\int ^{g_{2}}_{g_{1}}\frac{\zeta _{2}(w)}{w^{1-p}}\,dw-\zeta _{2} ( ( \frac{g_{1}^{p}+g_{2}^{p}}{2} ) ^{\frac{1}{p}} ) } =\frac{\zeta ^{\prime \prime }_{1}(\chi )}{\zeta ^{\prime \prime }_{2}(\chi )}, $$
(10)

provided that the denominators are non-zero.

Proof

Let

$$ H:= \biggl( \frac{g_{1}^{p}+g_{2}^{p}}{2} \biggr) ^{\frac{1}{p}} $$

and

$$ (T\zeta _{1}) (\lambda ):=\frac{p}{g_{2}^{p}-g_{1}^{p}} \int ^{g_{2}}_{g_{1}} \frac{\zeta _{1}(\lambda w+(1-\lambda )H)}{w^{1-p}}\,dw-\zeta _{1}(H), $$

where \(\lambda \in [0,1]\). Similarly, we can define \((T\zeta _{2})(\lambda )\).

Observe that

$$ (T\zeta _{1})^{\prime }(\lambda ):=\frac{p}{g_{2}^{p}-g_{1}^{p}} \int ^{g_{2}}_{g_{1}}(w-H) \frac{\zeta ^{\prime }_{1}(\lambda w+(1-\lambda )H)}{w^{1-p}}\,dw $$

and

$$ (T\zeta _{1})^{\prime \prime }(\lambda ):=\frac{p}{g_{2}^{p}-g_{1}^{p}} \int ^{g_{2}}_{g_{1}}(w-H)^{2} \frac{\zeta ^{\prime \prime }_{1}(\lambda w+(1-\lambda )H)}{w^{1-p}}\,dw. $$

Now consider the function \(Q(\lambda )\) defined by

$$ Q(\lambda )=(T\zeta _{2}) (1) (T\zeta _{1}) (\lambda )-(T\zeta _{1}) (1) (T \zeta _{2}) (\lambda ) $$

such that we have

$$ Q(0)=Q(1)=Q^{\prime }(0)=0. $$

Then, from two applications of mean value theorem, we find \(\upsilon \in G\) such that

$$ Q^{\prime \prime }(\upsilon )=0. $$

It implies

$$\begin{aligned} &\frac{p}{g_{2}^{p}-g_{1}^{p}} \int _{[g_{1},g_{2}]}(w-H)^{2}\bigl[(T\zeta _{2}) (1) \zeta _{1}^{\prime \prime }\bigl(w\upsilon -(1- \upsilon )H\bigr) \\ &\quad {} -(T\zeta _{1}) (1)\zeta _{2}^{\prime \prime } \bigl(w \upsilon -(1-\upsilon )H\bigr)\bigr]=0. \end{aligned}$$
(11)

For any fixed υ, the expression in the square brackets in (11) is a continuous function of w, so it vanishes. Corresponding to that value of w, we get a number

$$ \chi =w\upsilon +(1-\upsilon )H $$

such that

$$ (T\zeta _{2}) (1).\zeta _{1}^{\prime \prime }(\chi )-(T \zeta _{1}) (1).\zeta _{2}^{\prime \prime }( \chi )=0. $$

This gives equality (10). □

Corollary 3.2

If \(\frac{\zeta ^{\prime \prime }_{1}}{\zeta ^{\prime \prime }_{2}}\) is invertible, then we have

$$ \chi = \biggl( \frac{\zeta _{1}^{\prime \prime }(\chi )}{ \zeta _{2}^{\prime \prime }(\chi )} \biggr) ^{-1} \biggl( \frac{\frac{p}{g_{2}^{p}-g_{1}^{p}}\int ^{g_{2}}_{g_{1}}\frac{\zeta _{1}(w)}{w^{1-p}}\,dw -\zeta _{1} ( ( \frac{g_{1}^{p}+g_{2}^{p}}{2} ) ^{\frac{1}{p}} ) }{\frac{p}{g_{2}^{p}-g_{1}^{p}}\int ^{g_{2}}_{g_{1}}\frac{\zeta _{2}(w)}{w^{1-p}}\,dw-\zeta _{2} ( ( \frac{g_{1}^{p}+g_{2}^{p}}{2} ) ^{\frac{1}{p}} ) } \biggr) . $$
(12)

Corollary 3.3

By taking \(\zeta _{2}(w)={w}^{2}\) and \(\zeta _{1}=\zeta \) in Theorem 3.1, we have

$$\begin{aligned} \begin{aligned}[b] &\frac{p}{g_{2}^{p}-g_{1}^{p}} \int ^{g_{2}}_{g_{1}} \frac{\zeta (w)}{w^{1-p}}\,dw-\zeta \biggl( \biggl( \frac{g_{1}^{p}+g_{2}^{p}}{2} \biggr) ^{\frac{1}{p}} \biggr) \\ &\quad =\frac{\zeta ^{\prime \prime }(\chi )}{2} \biggl[ \frac{p}{g_{2}^{p}-g_{1}^{p}} \biggl( \frac{g_{2}^{p+2}-g_{1}^{p+2}}{p+2} \biggr) - \biggl( \frac{g_{1}^{p}+g_{2}^{p}}{2} \biggr) ^{\frac{2}{p}} \biggr] . \end{aligned} \end{aligned}$$
(13)

Cauchy type means for s-convex functions in Jensen’s sense

Here first we prove Jensen’s inequality for s-convex function.

Lemma 4.1

Let \(s\in (0,1]\) and \(\zeta :G\subset \mathbb{R}^{+}\rightarrow \mathbb{R}\) be an s-convex function. Let \(\sum^{n}_{1}r_{i}g_{i}\) be convex combinations of points \(g_{i}\in G\) with coefficients \(r_{i}\in [0,1]\). Then each s-convex function (in the first sense) satisfies the inequality

$$ \zeta \Biggl( \sum^{n}_{1}r_{i}g_{i} \Biggr) \leq \sum^{n}_{1}r_{i}^{s} \zeta (g_{i}), $$
(14)

where \(\sum^{n}_{1}r_{i}^{s}=1\).

Proof

We apply induction on the number of points in convex combination.

Basis step: for \(n=1\), equality (14) is true since

$$ \zeta (r_{1}g_{1})\leq r_{1}^{s} \zeta (g_{1}), $$

where \(r^{s}_{1}=1\) since \(r_{1}=1\).

Induction step: suppose that (14) holds for all convex combinations of points containing less than or equal to \(n-1\) points. Let \(r_{n}\neq 1\) and

$$ w=\sum_{1}^{n-1}\frac{r_{i}}{1-r_{n}}g_{i}, $$

where the sum \(\sum_{1}^{n-1} ( \frac{r_{i}}{1-r_{n}} ) g_{i}\in G\). Then, by induction hypothesis, we have

$$ \zeta (w)\leq \sum_{1}^{n-1} \biggl( \frac{r_{i}}{1-r_{n}} \biggr) ^{s} \zeta (g_{i}). $$
(15)

By using (3) and (15), we get

$$\begin{aligned} \begin{aligned}[b] \zeta \Biggl( \sum_{1}^{n}r_{i}g_{i} \Biggr) &=\zeta \bigl((1-r_{n})w+r_{n}g_{n} \bigr) \\ &\leq (1-r_{n})^{s}\zeta (w)+r_{n}^{s} \zeta (g_{n}) \\ &\leq (1-r_{n})^{s}\sum_{1}^{n-1} \biggl( \frac{r_{i}}{1-r_{n}} \biggr) ^{s}\zeta (g_{i})+r_{n}^{s} \zeta (g_{n}) \\ &=\sum_{1}^{n}r_{i}^{s} \zeta (g_{i}). \end{aligned} \end{aligned}$$
(16)

Thus we get (14). □

Remark 4.1

By taking \(s=1\) in Lemma 4.1 we can get Jensen’s inequality for convex function.

Now, by using the above lemma, we state and prove the following theorem, which gives the Cauchy type means for s-convex function.

Theorem 4.1

Let \(s\in (0,1]\) and \(r_{i}\in [0,1]\). Let \(\zeta _{1},\zeta _{2}\in C^{2}(G\subset [0,\infty ))\) be s-convex functions (in the first sense). Then there exist some \(\chi \in G\) such that the following equality holds:

$$ \frac{\sum^{n}_{1}r_{i}^{s}\zeta _{1}(g_{i})-\zeta _{1}(\sum^{n}_{1}r_{i}g_{i})}{\sum^{n}_{1}r_{i}^{s}\zeta _{2}(g_{i})-\zeta _{2}(\sum^{n}_{1}r_{i}g_{i})} =\frac{\zeta ^{\prime \prime }_{1}(\chi )}{\zeta ^{\prime \prime }_{2}(\chi )} $$
(17)

with each \(r_{i}\in [0,1]\) such that \(\sum^{n}_{1}r_{i}^{s}=1\) and provided that the denominators are non-zero.

Proof

Define

$$ H:=\sum^{n}_{1}r_{i}g_{i} $$

and

$$ (T\zeta _{1}) (\lambda ):=\sum^{n}_{1}r_{i}^{s} \zeta _{1}\bigl(\lambda g_{i}+(1- \lambda )H\bigr)-\zeta _{1}(H), $$

where \(\lambda \in [0,1]\). Accordingly, we can define \((T\zeta _{2})(\lambda )\).

Note that

$$ (T\zeta _{1})^{\prime }(\lambda ):=\sum ^{n}_{1}r_{i}^{s}(g_{i}-H) \zeta ^{\prime }_{1}\bigl( \lambda g_{i}+(1- \lambda )H\bigr) $$

and

$$ (T\zeta _{1})^{\prime \prime }(\lambda ):=\sum ^{n}_{1}r_{i}^{s}(g_{i}-H)^{2} \zeta ^{\prime \prime }_{1}\bigl(\lambda g_{i}+(1-\lambda )H\bigr). $$

Now consider the function \(Q(\lambda )\) defined by

$$ Q(\lambda )=(T\zeta _{2}) (1) (T\zeta _{1}) (\lambda )-(T\zeta _{1}) (1) (T \zeta _{2}) (\lambda ) $$

such that we have

$$ Q(0)=Q(1)=Q^{\prime }(0)=0. $$

Then, from two applications of mean value theorem, we find \(\upsilon \in G\) such that

$$Q^{\prime \prime }(\upsilon )=0. $$

It follows that

$$ \sum_{i=1}^{n}r_{i}^{s}(g_{i}-H)^{2} \bigl[(T\zeta _{2}) (1).\zeta ^{\prime \prime }_{1}\bigl( \upsilon g_{i} +(1-\upsilon )H\bigr) -(T\zeta _{1}) (1). \zeta ^{\prime \prime }_{2}\bigl( \upsilon g_{i} +(1- \upsilon )H\bigr)\bigr]=0. $$
(18)

For any fixed υ, the expression in the square brackets in (18) is a continuous function of \(g_{i}\), so it vanishes. Corresponding to that value of \(g_{i}\), we get a number

$$ \chi =\upsilon +(1-\upsilon )H, $$

so that

$$ (T\zeta _{2}) (1).\zeta ^{\prime \prime }_{1}(\chi )-(T \zeta _{1}) (1).\zeta ^{\prime \prime }_{2}( \chi )=0. $$

This gives equality (17). □

Corollary 4.2

Let \(s\in (0,1]\). Let \(\zeta _{1},\zeta _{2}\in C^{2}(G\subset [0,\infty ))\) be s-convex functions (in the first sense) such that \(\frac{\zeta ^{\prime \prime }_{1}}{\zeta ^{\prime \prime }_{2}}\) is invertible. Then there exist some \(\chi \in G\) such that the following equality holds:

$$ \chi = \biggl( \frac{\zeta ^{\prime \prime }_{1}}{\zeta ^{\prime \prime }_{2}} \biggr) ^{-1} \biggl( \frac{\sum^{n}_{1}r_{i}^{s}\zeta _{1}(g_{i})-\zeta _{1}(\sum^{n}_{1}r_{i}g_{i})}{\sum^{n}_{1}r_{i}^{s}\zeta _{2}(g_{i})-\zeta _{2}(\sum^{n}_{1}r_{i}g_{i})} \biggr), $$
(19)

with each \(r_{i}\in [0,1]\) such that \(\sum^{n}_{1}r_{i}^{s}=1\) and provided that the denominators are non-zero.

Corollary 4.3

Let \(s_{1},s_{2}\in (0,1)\). Let \(\zeta _{1},\zeta _{2}\in C^{2}((0,\infty ))\) be an \(s_{1}\)-convex function and an \(s_{2}\)-convex function (in the first sense), respectively, defined as \(\zeta _{1}(w)=w^{s_{1}}\) and \(\zeta _{2}(w)=w^{s_{2}}\). Then, from Theorem 4.1, we get

$$ \frac{\sum^{n}_{1}r_{i}^{s_{1}}(g_{i})^{s_{1}}- ( \sum^{n}_{1}r_{i}g_{i} ) ^{s_{1}}}{\sum^{n}_{1}r_{i}^{s_{2}}(g_{i})^{s_{2}}- ( \sum^{n}_{1}r_{i}g_{i} ) ^{s_{2}}} =\frac{s_{1}(s_{1}-1)}{s_{2}(s_{2}-1)}(\chi )^{s_{1}-s_{2}}. $$
(20)

Cauchy type means for s-convex functions in the Hermite–Hadamard sense

Drgomir and Fitzpatrick [7] gave the following result.

Theorem 5.1

Suppose that \(\zeta :[0,\infty )\rightarrow \mathbb{R}\) is an s-convex function in the first sense, where \(s\in (0,1)\), and let \(g_{1},g_{2}\in [0,\infty )\), \(g_{1}\leq g_{2}\). Then the following inequality holds:

$$ \zeta \biggl( \frac{g_{1}+g_{2}}{2} \biggr) \leq \frac{1}{g_{2}-g_{1}} \int ^{g_{2}}_{g_{1}}\zeta (w)\,dw\leq \frac{\zeta (g_{1})+s\zeta (g_{2})}{s+1}. $$
(21)

The above inequalities are sharp.

From inequality (21) we give the following result.

Theorem 5.2

Suppose that \(\zeta _{1},\zeta _{2}:[0,\infty )\rightarrow \mathbb{R}\) is an s-convex function in the first sense, where \(s\in (0,1)\), and let \(g_{1},g_{2}\in [0,\infty )\), \(g_{1}\leq g_{2}\). Let \(\zeta _{1},\zeta _{2}\in C^{2}([g_{1},g_{2}])\). Then there exist some \(\chi \in [g_{1},g_{2}]\) such that the following equality holds:

$$ \frac{\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}}\zeta _{1}(w)\,dw-\zeta _{1}(\frac{g_{1}+g_{2}}{2})}{\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}}\zeta _{2}(w)\,dw-\zeta _{2}(\frac{g_{1}+g_{2}}{2})} =\frac{\zeta ^{\prime \prime }_{1}(\chi )}{\zeta ^{\prime \prime }_{2}(\chi )}, $$
(22)

provided that the denominators are non-zero.

Proof

Let

$$ H:=\frac{g_{1}+g_{2}}{2} $$

and

$$ (T\zeta ) (\lambda ):=\frac{1}{g_{2}-g_{1}} \int ^{g_{2}}_{g_{1}}\zeta _{1}\bigl( \lambda w+(1-\lambda )H\bigr)\,dw-\zeta _{1}(H), $$

where \(\lambda \in [0,1]\). Accordingly, we can define \((T\zeta _{2})(\lambda )\).

We can have

$$ (T\zeta _{1})^{\prime }(\lambda ):=\frac{1}{g_{2}-g_{1}} \int ^{g_{2}}_{g_{1}}(w-H) \zeta _{1}^{\prime } \bigl(\lambda w+(1-\lambda )H\bigr)\,dw $$

and

$$ (T\zeta _{1})^{\prime \prime }(\lambda ):=\frac{1}{g_{2}-g_{1}} \int ^{g_{2}}_{g_{1}}(w-H)^{2} \zeta _{1}^{\prime \prime }\bigl(\lambda w+(1-\lambda )H\bigr)\,dw. $$

Now consider the function \(Q(\lambda )\) defined by

$$ Q(\lambda )=(T\zeta _{2}) (1) (T\zeta _{1}) (\lambda )-(T\zeta _{1}) (1) (T \zeta _{2}) (\lambda ) $$

such that we have

$$ Q(0)=Q(1)=Q^{\prime }(0)=0. $$

Then, from two applications of mean value theorem, we find \(\upsilon \in [g_{1},g_{2}]\) such that

$$ Q^{\prime \prime }(\upsilon )=0. $$

It implies

$$ \begin{aligned}[b] &\frac{1}{g_{2}-g_{1}} \int _{[g_{1},g_{2}]}(w-H)^{2}\bigl[(T\zeta _{2}) (1). \zeta _{1}^{\prime \prime }\bigl(w\upsilon -(1-\upsilon )H\bigr) \\ &\quad {}-(T\zeta _{1}) (1).\zeta _{2}^{\prime \prime } \bigl(w \upsilon -(1-\upsilon )H\bigr)\bigr]=0. \end{aligned} $$
(23)

For some fixed υ, the expression in the square brackets in (23) is a continuous function of w, so it vanishes. Corresponding to that value of w, we get a number

$$ \chi =w\upsilon +(1-\upsilon )H $$

such that

$$ (T\zeta _{2}) (1).\zeta _{1}^{\prime \prime }(\chi )-(T \zeta _{1}) (1).\zeta _{2}^{\prime \prime }( \chi )=0. $$

Thus we get (22). □

Corollary 5.3

If \(\frac{\zeta _{1}^{\prime \prime }}{\zeta _{2}^{\prime \prime }}\) is invertible, then we have

$$ \chi = \biggl( \frac{\zeta _{1}^{\prime \prime }(\chi )}{ \zeta _{2}^{\prime \prime }(\chi )} \biggr) ^{-1} \biggl( \frac{\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}}\varPsi _{1}(w)\,dw-\zeta _{1}(\frac{g_{1}+g_{2}}{2})}{\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}}\zeta _{2}(w)\,dw-\zeta _{2}(\frac{g_{1}+g_{2}}{2})} \biggr) . $$
(24)

Corollary 5.4

Let \(s_{1},s_{2}\in (0,1)\). By taking \(\zeta _{1}(w)=w^{s_{1}}\) and \(\zeta _{2}(w)={w}^{s_{2}}\), where \(w\in (0,\infty )\), in Theorem 5.2we have

$$ \frac{\frac{g_{2}^{s_{1}+1}-g_{1}^{s_{1}+1}}{(s_{1}+1)(g_{2}-g_{1})}-(\frac{g_{1}+g_{2}}{2})^{s_{1}}}{\frac{g_{2}^{s_{2}+1}-g_{1}^{s_{2}+1}}{(s_{2}+1)(g_{2}-g_{1})}-(\frac{g_{1}+g_{2}}{2})^{s_{2}}} =\frac{s_{1}(s_{1}-1)}{s_{2}(s_{2}-1)}(\chi )^{s_{1}-s_{2}}. $$
(25)

Now we define the following definition.

Definition 5.1

Let \(s\in (0,1)\) and \(g_{1},g_{2}\in [0,\infty )\), \(g_{1}\leq g_{2}\). Then quasi-arithmetic mean for the strictly monotonic function Φ defined on \([g_{1},g_{2}]\) is as follows:

$$ \widehat{M}_{\varPhi }(g_{1},g_{2})=\varPhi ^{-1} \biggl( \frac{1}{g_{2}-g_{1}} \int ^{g_{2}}_{g_{1}}\varPhi (w)\,dw-\varPhi \biggl( \frac{g_{1}+g_{2}}{2} \biggr) \biggr) . $$
(26)

Theorem 5.5

Let \(s\in (0,1)\) and \(g_{1},g_{2}\in [0,\infty )\), \(g_{1}\leq g_{2}\). Let \(\varPhi _{1},\varPhi _{2},\varPhi _{3}\in C^{2}([g_{1},g_{2}])\) be strictly monotonic real-valued functions. Then

$$ \frac{\varPhi _{1} ( \widehat{M}_{\varPhi _{1}}(g_{1},g_{2}) ) -\varPhi _{1} ( \widehat{M}_{\varPhi _{3}}(g_{1},g_{2}) ) }{\varPhi _{2} ( \widehat{M}_{\varPhi _{2}}(g_{1},g_{2}) ) -\varPhi _{2} ( \widetilde{M}_{\varPhi _{3}}(g_{1},g_{2}) ) } = \frac{\varPhi ^{\prime \prime }_{1}(\upsilon )\varPhi ^{\prime }_{3}(\upsilon )-\varPhi ^{\prime }_{1}(\upsilon )\varPhi ^{\prime \prime }_{3}(\upsilon )}{\varPhi ^{\prime \prime }_{2}(\upsilon )\varPhi ^{\prime }_{3}(\upsilon )-\varPhi ^{\prime }_{2}(\eta )\varPhi ^{\prime \prime }_{3}(\upsilon )} $$
(27)

for some υ, provided that the denominators are non-zero.

Proof

Let us choose functions \(\zeta _{1}=\varPhi _{1}\circ \varPhi ^{-1}_{3}\), \(\zeta _{2}=\varPhi _{2}\circ \varPhi ^{-1}_{3}\), \(w=\varPhi _{3}(w)\), and \(\frac{g_{1}+g_{2}}{2}=\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}} \varPhi _{3}(w)\,dw\) in Theorem 5.2, we observe that there exists some \(\upsilon \in [g_{1},g_{2}]\) such that

$$\begin{aligned} \begin{aligned}[b] &\frac{\varPhi _{1} ( \widehat{M}_{\varPhi _{1}}(g_{1},g_{2}) ) -\varPhi _{1} ( \widehat{M}_{\varPhi _{3}}(g_{1},g_{2}) ) }{\varPhi _{2} ( \widehat{M}_{\varPhi _{2}}(g_{1},g_{2}) ) -\varPhi _{2} ( \widehat{M}_{\varPhi _{3}}(g_{1},g_{2}) ) } \\ &\quad = \frac{\varPhi ^{\prime \prime }_{1}(\varPhi ^{-1}_{3}(\chi ))\varPhi ^{\prime }_{3}(\varPhi ^{-1}_{3}(\chi ))-\varPhi ^{\prime }_{1}(\varPhi ^{-1}_{3}(\chi ))\varPhi ^{\prime \prime }_{3}(\varPhi ^{-1}_{3}(\chi ))}{\varPhi ^{\prime \prime }_{2}(\varPhi ^{-1}_{3}(\chi ))\varPhi ^{\prime }_{3}(\varPhi ^{-1}_{3}(\chi ))-\varPhi ^{\prime }_{2}(\varPhi ^{-1}_{3}(\chi ))\varPhi ^{\prime \prime }_{3}(\varPhi ^{-1}_{3}(\chi ))}. \end{aligned} \end{aligned}$$
(28)

Then, by letting \(\varPhi ^{-1}_{3}(\chi )=\upsilon \), we notice that we have \(\upsilon \in [g_{1},g_{2}]\) such that

$$ \frac{\varPhi _{1} ( \widehat{M}_{\varPhi _{1}}(g_{1},g_{2}) ) -\varPhi _{1} ( \widehat{M}_{\varPhi _{3}}(g_{1},g_{2}) ) }{\varPhi _{2} ( \widehat{M}_{\varPhi _{2}}(g_{1},g_{2}) ) -\varPhi _{2} ( \widehat{M}_{\varPhi _{3}}(g_{1},g_{2}) ) } = \frac{\varPhi ^{\prime \prime }_{1}(\upsilon )\varPhi ^{\prime }_{3}(\upsilon )-\varPhi ^{\prime }_{1}(\upsilon )\varPhi ^{\prime \prime }_{3}(\upsilon )}{\varPhi ^{\prime \prime }_{2}(\upsilon )\varPhi ^{\prime }_{3}(\upsilon )-\varPhi ^{\prime }_{2}(\upsilon )\varPhi ^{\prime \prime }_{3}(\upsilon )}. $$
(29)

 □

Again from inequality (21) we have following result.

Theorem 5.6

Suppose that \(\zeta _{1},\zeta _{2}:[0,\infty )\rightarrow \mathbb{R}\) is an s-convex function in the first sense, where \(s\in (0,1)\), and let \(g_{1},g_{2}\in [0,\infty )\), \(g_{1}\leq g_{2}\). Let \(\zeta _{1},\zeta _{2}\in C^{2}([g_{1},g_{2}])\). Then there exist some \(\chi \in [g_{1},g_{2}]\) such that the following equality holds:

$$ \frac{\frac{\zeta _{1}(g_{1})+s\zeta _{1}(g_{2})}{s+1}-\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}}\zeta _{1}(w)\,dw}{\frac{\zeta _{2}(g_{1})+s\zeta _{2}(g_{2})}{s+1}-\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}}\zeta _{2}(w)\,dw} =\frac{\zeta ^{\prime \prime }_{1}(\chi )}{\zeta ^{\prime \prime }_{2}(\chi )}, $$
(30)

provided that the denominators are non-zero.

Proof

Consider the function

$$ (T\zeta _{1}) (w)=\frac{s\zeta _{1}(w)+\zeta _{1}(g_{1})}{s+1}(w-g_{1})- \int ^{w}_{g_{1}}\zeta _{1}(x)\,dx. $$
(31)

Similarly, we can define \(T\zeta _{2}(w)\).

Note that

$$ (T\zeta _{1})^{\prime }(w)=\frac{s\zeta _{1}^{\prime }(w)}{s+1}(w-g_{1})- \frac{\zeta _{1}(w)-\zeta _{1}(g_{1})}{s+1} $$
(32)

and

$$ (T\zeta _{1})^{\prime \prime }(w)=\frac{s\zeta _{1}^{\prime \prime }(w)}{s+1}(w-g_{1}). $$
(33)

We observe that

$$ (T\zeta _{1}) (g_{1})=(T\zeta _{1})^{\prime }(g_{1})=(T \zeta _{1})^{\prime \prime }(g_{1})=0. $$

Now we define \(D(w)\) as follows:

$$ D(w)=(T\zeta _{2}) (g_{2}) (T\zeta _{1}) (w)-(T\zeta _{1}) (g_{2}) (T \zeta _{2}) (w). $$
(34)

Then note that

$$ D(g_{1})=D^{\prime }(g_{2})=D^{\prime \prime }(g_{1})=D(g_{2})=0. $$

Thus, by application of the mean-value theorem, we get

$$ D^{\prime \prime }(\chi )=0 $$

for some \(\chi \in [g_{1},g_{2}]\). Consequently, this completes the proof of the theorem. □

Corollary 5.7

If \(\frac{\zeta _{1}^{\prime \prime }}{\zeta _{2}^{\prime \prime }}\) is invertible, then we have

$$ \chi = \biggl( \frac{\zeta _{1}^{\prime \prime }(\chi )}{ \zeta _{2}^{\prime \prime }(\chi )} \biggr) ^{-1} \biggl( \frac{\frac{\zeta _{1}(g_{1})+s\zeta _{1}(g_{2})}{s+1}-\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}} \zeta _{1}(w)\,dw}{\frac{\zeta _{2}(g_{1})+s\zeta _{2}(g_{2})}{s+1}-\frac{1}{g_{2}-g_{1}}\int ^{g_{2}}_{g_{1}} \zeta _{2}(w)\,dw} \biggr). $$
(35)

Corollary 5.8

Let \(s_{1},s_{2}\in (0,1)\). By taking \(\zeta _{1}(w)=w^{s_{1}}\) and \(\zeta _{2}(w)={w}^{s_{2}}\), where \(w\in (0,\infty )\), in Theorem 5.6, we have

$$ \frac{(g_{1}^{s_{1}}+s_{1}g_{2}^{s_{1}})- ( \frac{g_{2}^{s_{1}+1}-g_{1}^{s_{1}+1}}{g_{2}-g_{1}} ) }{(g_{1}^{s_{2}}+s_{2}g_{2}^{s_{2}})- ( \frac{g_{2}^{s_{2}+1}-g_{1}^{s_{2}+1}}{g_{2}-g_{1}} ) } =\frac{s_{1}(s_{1}-1)(s_{2}+1)}{s_{2}(s_{2}-1)(s_{1}+1)}(\chi )^{s_{1}-s_{2}}. $$
(36)

Conclusion

In Sect. 2, we proved Cauchy type mean for p-convex functions. In Sect. 3, Cauchy type theorem in the Hermite–Hadamard sense was obtained for p-convex functions. In Sect. 4, we proved Jensen’s inequality for s-convex functions in the first sense, and then a Cauchy type theorem was obtained. In Sect. 5, a Cauchy type theorem in the Hermite–Hadamard sense was obtained for s-convex functions in the first sense.

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The data and material used to support the findings of this study are included within the article.

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Acknowledgements

We thank the anonymous referees and editor for their careful reading of the manuscript and many insightful comments to improve the results.

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This research article is supported by the National University of Sciences and Technology(NUST), Islamabad, Pakistan.

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Mehreen, N., Anwar, M. Cauchy type means for some generalized convex functions. J Inequal Appl 2021, 114 (2021). https://doi.org/10.1186/s13660-021-02647-2

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Keywords

  • Cauchy mean value theorem
  • Jensen’s inequality
  • Hermite–Hadamard inequality
  • p-convex function
  • s-convex function in the first sense