Refinements of Jensen’s and McShane’s inequalities with applications

1.   Introduction

It is well-known that the Jensen inequality ^[1,2,3] for convex function is one of the most famous and important inequalities in the whole theory of inequalities. Recently, the generalizations and improvements for the Jensen inequality have been the subject of much research, it has been generalized to the $s$-convex ^[4], co-ordinate convex ^[5], $\phi$-convex ^[6], $\alpha(x)$-convex ^[7] and strongly convex functions ^[8]. It is worth noting that it is closely related to many other important inequalities such as Cauchy-Schwarz inequalities ^[9], Ostrowski inequalities ^[10], Minkowski inequalities ^[11], Hermite-Hadamard inequalities ^{[12,13,14,15,16,17]}, Bessel function inequalities ^[18], Petrović inequalities ^[19], Pólya-Szegö inequalities ^[20], exponentially convex inequalities ^[21], integral inequalities ^{[22,23,24,25,26,27]}, mean value inequalities ^{[28,29,30,31,32]}, gamma function inequalities ^[33], generalized convex functions inequalities ^[34], generalized trigonometric functions inequalities ^[35] and so on. The Jensen inequality can be stated as follows.

The inequality

holds for all $y_j\in I$ and $\zeta_j > 0$ $(j = 1, 2, \dots, n)$ if $\psi: I\rightarrow\mathbb{R}$ is a convex function.

The Jensen inequality (1.1) has wild applications in many fields of natural sciences, for example, in optimization theory, statistics, information theory and financial economics ^[36,37,38].

The main focus of this article is to present several refinements of the generalized Jensen's inequality for the isotonic linear functionals. Before giving Jessen's and McShane's results, we consider the following hypothesis and recall a definition.

Hypothesis $\textbf{H}$: Suppose that $\mathcal{M}$ is a non empty set and $\mathcal{L}$ is the class of real-valued functions $f : \mathcal{M}\rightarrow \mathbb{R}$ such that

(ⅰ) $\alpha_{1}g_{1}+\alpha_{2}g_{2}\in\mathcal{L}$ if $g_{1}, g_{2}\in\mathcal{L}$ and $\alpha_{1}, \alpha_{2}\in\mathbb{R}$;

(ⅱ) $g\in\mathcal{L}$ if $g(z) = 1$ for all $z\in\mathcal{M}$.

Definition 1.1. The functional $\mathcal{G}: \mathcal{L}\rightarrow\mathbb{R}$ is said to be an isotonic linear functional if it satisfies the following two conditions:

(ⅰ) $\mathcal{G}(\alpha_1 g_1+\alpha_2 g_2) = \alpha_1 \mathcal{G}(g_1)+\alpha_2 \mathcal{G}(g_2)$ for $g_1, g_2\in \mathcal{L}, \alpha_1, \alpha_2\in \mathbb{R}$;

(ⅱ) $g\in \mathcal{L}, g(t)\geq0$ on $\mathcal{M}\Rightarrow \mathcal{G}(g)\geq0$.

In 1931, Jessen ^[39] constructed the functional version of the Jensen's inequality for convex functions with one variable. In the following Theorem 1.1, we present an weighted version of the Jessen's inequality.

Theorem 1.1. Let the hypothesis $\textbf{H}$ be true, $\mathcal{G}:\mathcal{L}\rightarrow\mathbb{R}$ be an isotonic linear functional and $\psi:[a, b]\rightarrow\mathbb{R}$ be a continuous convex function. Then for all $f, \vartheta\in \mathcal{L}$ such that $\vartheta\psi(f), \vartheta f\in \mathcal{L}$ and $\mathcal{G}(\vartheta) > 0$, we have $\frac{\mathcal{G}(\vartheta f)}{\mathcal{G}(\vartheta)}\in [a, b]$ and

Inequality (1.2) has been applied to obtain the monotonicity of generalized means. We present those results for means as follows.

Let $r\in\mathbb{R}$. Then the generalization of the classical mean $M_r(\vartheta, f; \mathcal{G})$ for isotonic functional $\mathcal{G}$ is defined by

where $f(x) > 0$ for $x\in \mathcal{M}$, $\vartheta, \vartheta f^{r}\in \mathcal{L}$ for $r\in\mathbb{R}$, $\vartheta \log f \in \mathcal{L}$ and $\mathcal{G}(\vartheta) > 0$.

The following Theorem 1.2 for the monotonicity of the above generalized mean can be found in the literature ^[40].

Theorem 1.2. Let the hypothesis $\textbf{H}$ be true, $f, \vartheta$ be the functions defined on $\mathcal{M}$ such that $f, \vartheta, f^{r}, \vartheta f^{r}\in \mathcal{L}$ $(r\in\mathbb{R})$ and $f(x) > 0$ for $x\in \mathcal{M}$, and $\mathcal{G}$ be an isotonic linear functional defined on $\mathcal{L}$ such that $\mathcal{G}(\vartheta) > 0$. Then the inequality

holds for all $p\leq l$.

Let $\vartheta, h:[a, b]\rightarrow \mathbb{R}$ be the functions such that $h$ is strictly monotone and continuous, and $\vartheta h(f)\in \mathcal{L}$ for $f\in \mathcal{L}$ with $f(x)\in[a, b]$ and $\mathcal{G}(\vartheta) > 0$. Then the generalized quasi arithmetic mean is defined by

The following Theorem 1.3 on the monotonicity of the generalized quasi arithmetic mean is given in ^[40].

Theorem 1.3. Let the above hypotheses hold and $g:[a, b]\rightarrow\mathbb{R}$ be a strictly monotone and continuous function such that $\vartheta g(f)\in \mathcal{L}$ for $f\in \mathcal{L}$ with $f(x)\in[a, b]$ and $\mathcal{G}(\vartheta) > 0$. Then one has

if $g\circ h^{-1}$ is a convex function.

In 1937, McShane ^[41] extended the above functional version of Jensen's inequality from convex functions with one variable to the convex functions with several variables. The following Theorem 1.4 is a weighted version of McShane's result.

Theorem 1.4. Let the hypothesis $\textbf{H}$ be true, $\mathcal{G}:\mathcal{L}\rightarrow \mathbb{R}$ be an isotonic linear functional, $C$ be a convex closed subset of $\mathbb{R}^{n}$, $\psi$ be a convex and continuous function defined on $C$, $\phi_1(x), \phi_2(x), \dots, \phi_n(x), \vartheta (x)$ be the functions from $\mathcal{L}$ such that $\boldsymbol{\phi}(x) = (\phi_1(x), \phi_2(x), \dots, \phi_n(x))\in C$ for all $x\in \mathcal{M}$, $\vartheta \psi(\boldsymbol{\phi}(x)), \vartheta \phi_i\in \mathcal{L}$ $(i = 1, 2, \dots, n)$ and $\mathcal{G}(\vartheta) > 0$. Then one has

The following generalization of Beck's inequality can be found in the literature ^[40] by using McShane's inequality.

Theorem 1.5. Let the hypothesis $\textbf{H}$ be valid, $\mathcal{G}:\mathcal{L}\rightarrow\mathbb{R}$ be an isotonic linear functional, $\psi_i:I_i\rightarrow\mathbb{R}$ $(i = 1, 2, \dots, n)$ be continuous and strictly monotonic, $\tau:I\rightarrow \mathbb{R}$ be continuous and increasing, and $g_1, g_2, \dots, g_n:\mathcal{M}\rightarrow \mathbb{R}$ and $\psi:I_1\times I_2\times\cdots\times I_n\rightarrow \mathbb{R}$ be the real-valued functions such that $g_1(\mathcal{M})\subset I_1$, $g_2(\mathcal{M})\subset I_2$, $\dots$, $g_n(\mathcal{M})\subset I_n$, $\psi_1(g_1), \psi_2(g_2), \dots, \psi_n(g_n), \tau(\psi(g_1, g_2, \dots, g_n)), \vartheta \in \mathcal{L}$ and $\mathcal{G}(\vartheta) > 0$. Then the inequality

holds if the function $H$ defined by $H(s_1, s_2, \dots, s_n) = -\tau(\psi(\psi_1^{-1}(s_1), \psi_2^{-1}(s_2), \dots, \psi_n^{-1}(s_n))$ is convex.

Remark 1.1. It is important to note that Beck ^[42] gave the special case of Theorem 1.5 for discrete functionals with $n = 2$.

The main purpose of the article is to refine the Jessen's and McShane's inequalities associated to certain functions from the linear functions space, improve the generalized means, Hölder and McShane's inequalities, and generalize the Jessen's and McShane's inequalities containing $n$ certain functions.

2.   Refinement of Jessen's inequality with applications

We first present a refinement of the Jessen's inequality.

Theorem 2.1. Under the assumptions of Theorem 1.1, if $u, v\in \mathcal{L}$ such that $u(t)+v(t) = 1$ for $t\in\mathcal{M}$ and $u\vartheta f, v\vartheta f, u\vartheta, v\vartheta \in \mathcal{L}$ with $\mathcal{G}(u\vartheta), \mathcal{G}(v\vartheta) > 0$, then

Proof. It follows from $u(t)+v(t) = 1$ for $t\in \mathcal{M}$ and the linearity of $\mathcal{G}$ that

and

Making use of the convexity of $\psi$ on the right hand side of inequality (2.2) we obtain

Applying Jessen's inequality (1.2) on both sides of inequality (2.3), we have

Therefore, inequality (2.1) follows from inequalities (2.3) and (2.4).

From Theorem 2.1 we get Corollary 2.1, which is the refinement of inequality (1.4).

Corollary 2.1. Let the hypothesis $\textbf{H}$ be valid, $f, \vartheta, u, v$ be the functions defined on $\mathcal{M}$ such that $f, \vartheta, u, v, u\vartheta, v\vartheta, u\vartheta f^{r}, v\vartheta f^{^{r}}\vartheta f^{r}\in\mathcal{L}$ $(r\in\mathbb{R})$ and $f(x) > 0$ for $x\in \mathcal{M}$, $\mathcal{G}$ be an isotonic linear functional on $\mathcal{L}$ such that $\mathcal{G}(\vartheta), \mathcal{G}(u\vartheta), \mathcal{G}(v\vartheta) > 0$, $u(x)+v(x) = 1$ for $x\in \mathcal{M}$, $p, l\in\mathbb{R}$ with $p\leq l$. Then one has

for $l\neq 0$,

for $p\leq 0$,

for $p\neq 0$, and

for $l\geq 0$.

Proof. Let $p, l\neq 0$. Then using (2.1) for $\psi(z) = z^{\frac{l}{p}}$ $(z > 0)$, $f\rightarrow f^{p}$ and taking the power $\frac{1}{l}$ we get inequality (2.5).

Next, we prove inequality (2.6) by taking limit $l\rightarrow 0$ in (2.5). Let

Then it follows from linearity of $\mathcal{G}$ and $u+v = 1$ that

Let

Then

Taking $l\rightarrow 0$ and using l'Hôpital rule we obtain

that is

Taking $l\rightarrow 0$ in (2.5) and using (2.9) we obtain the desired inequality (2.6).

Similarly utilizing inequality (2.1) for $\psi(z) = z^{\frac{p}{l}}$, $(z > 0, p, l\neq0)$, $f\rightarrow f^{l}$ and taking power $\frac{1}{p}$ we get (2.7). Taking $p\rightarrow0$ in (2.7) leads to (2.8)

As an application of Theorem 2.1, we derive an refinement of inequality inequality (1.5).

Corollary 2.2. Let all the above hypotheses hold, $g:[a, b]\rightarrow \mathbb{R}$ be a strictly monotone and continuous function such that $\vartheta g(f)\in \mathcal{L}$ for $f\in \mathcal{L}$ with $f(x)\in[a, b]$, $u, v\in \mathcal{L}$ such that $u(x)+v(x) = 1$ for $x\in \mathcal{M}$ and $\mathcal{G}(\vartheta), \mathcal{G}(u\vartheta), \mathcal{G}(v\vartheta) > 0$. Then the inequality

holds if $g\circ h^{-1}$ is a convex function.

Proof. Inequality (2.10) can be derived by use of inequality (2.1) for $f\rightarrow h\circ f$ and $\psi\rightarrow g\circ h^{-1}$.

As applications of Theorem 2.1, we present the refinements of the Hölder inequality in the following Corollaries 2.3 and 2.4.

Corollary 2.3. Let the hypothesis $\textbf{H}$ be valid, $\mathcal{G}:\mathcal{L}\rightarrow \mathbb{R}$ be an isotonic linear functional, $r_{1} > 1$, $r_2 = \frac{r_{1}}{r_{1}-1}$, $u, v, w, g_1$ and $g_2$ be non-negative functions defined on $\mathcal{M}$ such that $wg_1^{r_1}, wg_2^{r_{2}}, uwg_2^{r_{2}}, vwg_2^{r_{2}}, uwg_1g_2, vwg_1g_2, wg_1g_2\in\mathcal{L}$ and $u(x)+v(x) = 1$ for $x\in \mathcal{M}$. Then

In the case of $0 < r_{1} < 1$ and $r_{2} = \frac{r_{1}}{r_{1}-1}$ with $\mathcal{G}(wg^{r_{2}}_2) > 0$ or $r_{1} < 0$ and $\mathcal{G}(wg^{r_{1}}_{1}) > 0$, we have

Proof. Assume that $\mathcal{G}(wg^{r_{2}}_2) > 0$. Since $wg_2^{r_{2}}g_{1}g_2^{\frac{-r_{2}}{r_{1}}} = wg_1g_2\in \mathcal{L}$ and $wg_{2}^{r_{2}}g^{r_1}_{1}g_{2}^{-r_{2}} = wg^{r_1}_{1}\in \mathcal{L}$, therefore by using Theorem 2.1 for $\psi(z) = z^{r_{1}}$ $(z > 0, r_{1} > 1)$, $\vartheta = wg_2^{r_{2}}$, $f = g_{1}g_2^{\frac{-r_{2}}{r_{1}}}$, we obtain inequality (2.11). If $\mathcal{G}(wg^{r_{1}}_{1}) > 0$, then applying the same procedure but taking $r_{1}, r_{2}, g_{1}, g_2$ instead of $r_{2}, r_{1}, g_2, g_{1}$, we also obtain inequality (2.11).

If $\mathcal{G}(wg^{r_{2}}_2) = \mathcal{G}(wg^{r_{1}}_{1}) = 0$, then as we know that

It gives that $\mathcal{G}(wg_{1}g_2) = 0$. The proof for the case $r_{1} > 1$ is completed.

If $0 < r_{1} < 1$, then $M = \frac{1}{r_{1}} > 1$, and the desired result can be obtained by using (2.11) for $M$, $N = (1-r_{1})^{-1}$, $\overline{g}_{1} = (g_{1}g_2)^{r_{1}}$, and $\overline{g}_2 = g^{-r_{1}}_2$ instead of $r_{1}$, $r_{2}$, $g_{1}$ and $g_2$,

Finally, if $r_{1} < 0$, then $0 < r_{2} < 1$, we may use the similar arguments with $r_{1}$, $r_{2}$, $g_{1}$, $g_2$ replaced by $r_{2}$, $r_{1}$, $g_2$, $g_{1}$ provided that $\mathcal{G}(wg^{r_{1}}_{1}) > 0$ to get the desired result.

Corollary 2.4. Let the hypothesis $\textbf{H}$ be true, $\mathcal{G}:\mathcal{L}\rightarrow\mathbb{R}$ be an isotonic linear functional, $r_{1} > 1$, $r_2 = \frac{r_{1}}{r_{1}-1}$, $u, v, w, g_1$ and $g_2$ be non-negative functions defined on $\mathcal{M}$ such that $wg_1^{r_1}, wg_2^{r_{2}}, uwg_2^{r_{2}}, vwg_2^{r_{2}}, uwg_1g_2, vwg_1g_2, wg_1g_2\in \mathcal{L}$ and $u(x)+v(x) = 1$ for $x\in \mathcal{M}$. Then

In the case of $0 < r_{1} < 1$ and $r_{2} = \frac{r_{1}}{r_{1}-1}$ with $\mathcal{G}(wg^{r_{2}}_{2}) > 0$ or $r_{1} < 0$ and $\mathcal{G}(wg^{r_{1}}_{1}) > 0$, we get

Proof. If $\mathcal{G}(wg^{r_{2}}_2) > 0$, then let $\psi(z) = -z^{\frac{1}{r_{1}}}$ $(z > 0, r_{1} > 1)$, we clearly see that the function $\psi$ is convex. Therefore, using Theorem 2.1 for $\psi(z) = -z^{\frac{1}{r_{1}}}$, $\vartheta = w g^{r_{2}}_2$ and $f = g^{r_{1}}_{1}g^{-r_{2}}_2$ we obtain (2.14). If $\mathcal{G}(wg^{r_{1}}_{1}) > 0$, then applying the same procedure but taking $r_{1}$, $r_{2}$, $g_{1}$ and $g_2$ instead of $r_{2}$, $r_{1}$, $g_2$ and $g_{1}$ we also get (2.14).

For the case of $\mathcal{G}(wg^{r_{2}}_2) = \mathcal{G}(wg^{r_{1}}_{1}) = 0$, we can complete the proof by use of the similar argument as in the proof of Corollary 2.3.

If $0 < r_{1} < 1$, then $M = \frac{1}{r_{1}} > 1$ and applying (2.14) for $M$ and $N = (1-r_{1})^{-1}$, $\overline{g}_{1} = (g_{1}g_2)^{r_{1}}$ and $\overline{g}_2 = g^{-r_{1}}_2$ instead of $r_{1}$, $r_{2}$, $g_{1}$ and $g_2$ we get (2.15).

Finally, if $r_{1} < 0$, then $0 < r_{2} < 1$, and we can get the desired result if we use the same method as above but instead $r_{1}$, $r_{2}$, $g_{1}$ and $g_2$ by $r_{2}$, $r_{1}$, $g_2$ and $g_{1}$ provided that $\int_{a}^{b}w(\varrho)g^{r_{1}}_{1}(\varrho)d\varrho > 0$.

Remark 2.1. In the above results if we consider $\mathcal{G}(\vartheta g) = \int_{a}^{b}p(x)f(x)dx$ and $\mathcal{G}(\vartheta) = \int_{a}^{b}p(x)dx$, then we obtain the results presented in ^[43], where $p, g:[a, b]\rightarrow\mathbb{R}$ are integrable functions with $p(x) > 0$ for $x\in[a, b]$.

3.   Refinement of McShane's inequality with applications

We begin this section by giving a refinement of the McShane's inequality.

Theorem 3.1. Under the assumptions of Theorem 1.4, if $u, v\in \mathcal{L}$ such that $u(x)+v(x) = 1$ for $x\in \mathcal{M}$ and $\vartheta\psi(\boldsymbol{\phi}(x)), u\vartheta\phi_i, v\vartheta\phi_i, u\vartheta, v\vartheta\in\mathcal{L}$ $(i = 1, 2, \dots, n)$ with $\mathcal{G}(u\vartheta), \mathcal{G}(v\vartheta) > 0$, then

Proof. It follows from the convexity, (1.7), $u(x)+v(x) = 1$ for $x\in\mathcal{M}$ and the linearity of $\mathcal{G}$ that

The following Theorem 3.2 provides a refinement of the generalized Beck's inequality (1.8).

Theorem 3.2. Let the hypothesis $\textbf{H}$ be valid, $\mathcal{G}:\mathcal{L}\rightarrow\mathbb{R}$ be an isotonic linear functional, $\psi_i: I_i\rightarrow \mathbb{R}$ $(i = 1, 2, \dots, n)$ be continuous and strictly monotonic, $\tau:I\rightarrow \mathbb{R}$ be a continuous and increasing function, $g_1, g_2, \cdots, g_n :\mathcal{M}\rightarrow\mathbb{R}$, $\psi:I_1\times I_2\times\cdots\times I_n\rightarrow \mathbb{R}$ be the real-valued functions such that $g_1(\mathcal{M})\subset I_1$, $g_2(\mathcal{M})\subset I_2$, $\cdots$, $g_n(\mathcal{M})\subset I_n$, $\psi_1(g_1)$, $\psi_2(g_2)$, $\cdots$, $\psi_n(g_n)$, $\tau(\psi(g_1, g_2, \dots, g_n))$, $u, v, \vartheta, u\vartheta, v\vartheta\in\mathcal{L}$, $u(x)+v(x) = 1$ for $x\in \mathcal{M}$ and $\mathcal{G}(\vartheta), \mathcal{G}(u\vartheta), \mathcal{G}(v\vartheta) > 0$. Then

if the function $H$ defined by $H(s_1, s_2, \cdots, s_n) = -\tau(\psi(\psi_1^{-1}(s_1), \psi_2^{-1}(s_2), \cdots, \psi_n^{-1}(s_n))$ is convex.

Proof. Applying Theorem 3.2 for the function $H$ instead of $\psi$, we obtain

Let $\phi_i = \psi_i(g_i)$ $(i = 1, 2, \dots, n)$. Then (3.3) becomes

which is equivalent to (3.2).

A refinement of the Beck's inequality is given in the following Corollary 3.1.

Corollary 3.1. Under the assumptions of Theorem 3.2 for $n = 2$, we have the following inequalities

if the function $H$ defined by $H(s_1, s_2) = -\tau(\psi(\psi_1^{-1}(s_1), \psi_2^{-1}(s_2))$ is convex.

Next, we discuss some particular cases of Corollary 3.1.

Corollary 3.2. Let all the assumptions of Theorem 3.2 hold for $n = 2$ with $\psi(z_1, z_2) = z_1+z_2$, and $\psi_1, \psi_2, \tau$ be the twice continuous differentiable functions such that $\psi^{\prime}_1$, $\psi^{\prime}_2$, $\tau^{\prime}$, $\psi^{\prime\prime}_1$, $\psi^{\prime\prime}_2$ and $\tau^{\prime\prime}$ are positive. Then the inequality

holds if $G(z_1) + H(z_2) \leq K(z_1 + z_2)$, where $G = \frac{\psi_1^{\prime}}{\psi_1^{\prime\prime}}$, $H = \frac{\psi_2^{\prime}}{\psi_2^{\prime\prime}}$ and $K = \frac{\tau^{\prime}}{\tau^{\prime\prime}}$.

Proof. The idea of the proof is similar to the proof of Corollary 3.2 given in ^[44].

Similar to the idea of Corollary 3.2, we state the following Corollary 3.3.

Corollary 3.3. Let all the assumptions of Theorem 3.2 hold for $n = 2$ with $\psi(z_1, z_2) = z_1z_2$, $\psi_1, \psi_2, \tau$ be the twice continuous differentiable functions and $L_1(z) = \frac{\psi_1^{\prime}(z)}{\psi_1^{\prime}(z)+z\psi_1^{\prime\prime}(z)}$, $L_2(z) = \frac{\psi_2^{\prime}(z)}{\psi_2^{\prime}(z)+z\psi_2^{\prime\prime}(z)}$, $L_3(z) = \frac{\tau^{\prime}(z)}{\tau^{\prime}(z)+z\tau^{\prime\prime}(z)}$ such that $\psi_1^{\prime}$, $\psi_2^{\prime}$, $\tau^{\prime}$, $\tau^{\prime\prime}$, $\psi_1^{\prime\prime}$, $\psi_2^{\prime\prime}$, $L_1$, $L_2$, $L_3$ are positive. Then the inequality

holds if $L_1(z_1)+L_2(z_2)\leq L_3(z_1z_2)$.

4.   Further generalizations

The following Theorem 4.1 provides further generalization for the refinement of the Jessen's inequality associated to $n$ certain functions.

Theorem 4.1. Let all the assumptions of Theorem 1.1 hold, $u_l\in\mathcal{L}$ such that $\sum\limits_{l = 1}^{n}u_l = 1$ and $u_l\vartheta f, u_l\vartheta\in\mathcal{L}$ with $\mathcal{G}(u_l\vartheta) > 0$ for all $l\in\{1, 2, \cdots, n\}$, and $\mathcal{S}_1, \mathcal{S}_2\subset\{1, 2, \dots, n\}$ such that $\mathcal{S}_1$ and $\mathcal{S}_2$ are non empty, $\mathcal{S}_1\cap\mathcal{S}_2 = \emptyset$ and $\mathcal{S}_1\cup\mathcal{S}_2 = \{1, 2, \dots, n\}$. Then one has

Proof. It follows from the linearity of $\mathcal{G}$, $\sum\limits_{l = 1}^{n}u_l = 1$, Jessen's inequality and the definition of the convexity that

and

Similar to the above Theorem 4.1, in the following Theorem 4.2 we give further generalization of the McShane's inequality.

Theorem 4.2. Let all the assumptions of Theorem 1.4 hold, $u_l\in \mathcal{L}$ such that $\sum\limits_{l = 1}^{n}u_l = 1$ and $u_l\vartheta\psi(\boldsymbol{\phi}(x)$), $v_l\vartheta\psi(\boldsymbol{\phi}(x))$, $u_l\vartheta\phi_l$, $u_l\vartheta\in\mathcal{L}$ with $\mathcal{G}(u_l\vartheta) > 0$ for all $l\in\{1, 2, \cdots, n\}$, and $\mathcal{S}_1$ and $\mathcal{S}_2$ are non empty and disjoint subsets of $\{1, 2, \cdots, n\}$ such that $\mathcal{S}_1\cup\mathcal{S}_2 = \{1, 2, \cdots, n\}$. Then

Proof. It follows from (1.7), $\sum_{l = 1}^{n}u_l(x) = 1$ for $x\in \mathcal{M}$, the linearity of $\mathcal{G}$ and the definition of the convexity that

Remark 4.1. Analogously as in the section 2, we may give some applications of Theorems 4.1 and 4.2.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The research is supported by the Natural Science Foundation of China (Grant No. 61772025).

Conflict of interest

The authors declare that they have no competing interests.