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.

1 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 [1–5, 18–21]. Also see [8–11, 15–17] 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 $r_{1}^{s}+r_{2}^{s}=1$.

2 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). □

3 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)

4 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)

5 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

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

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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:

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

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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

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

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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

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

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

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

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 □

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:

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

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provided that the denominators are non-zero.

Proof

Consider the function

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

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

Note that

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

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and

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

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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:

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

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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

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

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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

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

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6 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.