New improvements of Jensen's type inequalities via 4-convex functions with applications (Q2222836)

Assume that \(T:[\rho_1, \rho_2]\to\mathbb{R}\) is a convex function, \(s_k\in[\rho_1, \rho_2],~ u_k\ge 0~(k=1, 2, \cdots, m)\) with \(U_m=\sum_{k=1}^{m}u_k>0\), the Jensen inequality state that \[ T\left(\frac{1}{U_m}\sum_{k=1}^{m}u_ks_k\right) \le \frac{1}{U_m}\sum_{k=1}^mu_kT(s_k)\,. \] The Jensen inequality extended the continuous case as follows \[ T\left(\frac{1}{\int_{\rho_1}^{\rho_2}p(x)dx}\int_{\rho_1}^{\rho_2}xp(x)dx\right)\le \frac{1}{\int_{\rho_1}^{\rho_2}p(x)dx}\int_{\rho_1}^{\rho_2}T(x)p(x)dx\,, \] where \(p(x)\) is a nonnegative weight function with \(\int_{\rho_1}^{\rho_2}p(x)dx>0\). In this paper under review, the authors present that if \(T\in C^2[\rho_1, \rho_2]\) is a 4-convex function and \(s_k\in[\rho_1, \rho_2]\), \(u_k\in\mathbb{R}~(k=1, \cdots, m)\) with \(U_m=\sum_{k=1}^{m}u_k\neq 0\) and \[ \frac{1}{U_m}\sum_{k=1}^mu_kG_i(s_k, x)-G_i\left(\frac{1}{U_m}\sum_{k=1}^mu_ks_kx\right)\ge 0\qquad(i=1, 2, 3, 4, 5) \] where \begin{align*} G_1(z, x)=&\begin{cases} \rho_1-x &; \rho_1 \le x \le z\\ \rho_1-z &; z \le x \le \rho_2 \end{cases}\\ G_2(z, x)=&\begin{cases} z- \rho_2 &; \rho_1 \le x \le z\\ x-\rho_2 &; z \le x \le \rho_2 \end{cases}\\ G_3(z, x)=&\begin{cases} z- \rho_1 &; \rho_1 \le x \le z\\ x-\rho_1 &; z \le x \le \rho_2 \end{cases}\\ G_4(z, x)=&\begin{cases} \rho_2-x &; \rho_1 \le x \le z\\ \rho_2-z &; z \le x \le \rho_2 \end{cases}\\ G_5(z, x)=&\begin{cases} \frac{(z- \rho_2)(x-\rho_1)}{\rho_2-\rho_1} &; \rho_1 \le x \le z\\ \frac{(x- \rho_2)(z-\rho_1)}{\rho_2-\rho_1} &; z \le x \le \rho_2 \end{cases} \end{align*} then \begin{align*} \frac{1}{U_m}&\sum_{k=1}^mu_kT(s_k)-T\left(\frac{1}{U_m}\sum_{k=1}^mu_ks_k\right)\\ \le&\frac{T''(\rho_2)-T''(\rho_1)}{6(\rho_2-\rho_1)}\left(\frac{1}{U_m}\sum_{k=1}^mu_ks_k^3-\left(\frac{1}{U_m}\sum_{k=1}^mu_ks_k\right)^3\right)\\ &+\frac{\rho_2T''(\rho_1)-\rho_1T''(\rho_2)}{2(\rho_2-\rho_1)}\left(\frac{1}{U_m}\sum_{k=1}^mu_ks_k^2-\left(\frac{1}{U_m}\sum_{k=1}^mu_ks_k\right)^2\right)\,. \end{align*} Also, the authors give an improvement of Jensen's inequality. In continuation, the authors propound the integral version of the above inequality.