# 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.

## Availability of data and materials

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.

## Funding

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

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Both authors contributed equally to this work. Both authors read and approved the final manuscript.

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Correspondence to Naila Mehreen.

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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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• DOI: https://doi.org/10.1186/s13660-021-02647-2

### Keywords

• Cauchy mean value theorem
• Jensen’s inequality