In this article, we present some new improvements of Jensen’s type inequalities via 4-convex and Green functions. These improvements are demonstrated in discrete as well as in integral versions. The aforesaid results enable us to give some improvements of Jensen’s and the Jensen–Steffensen inequalities. Also, we present some improvements of the reverse Jensen’s and the Jensen–Steffensen inequalities. Then as consequences of the improved Jensen’s inequality, we deduce some new bounds for the power, geometric and quasi-arithmetic means, also obtain bounds for the Hermite–Hadamard gap and improvements of the Hölder inequality. Finally as applications of the improved Jensen’s inequality, we present some new bounds for various divergences and Zipf–Mandelbrot entropy.

在本文中，我们通过4-凸和Green函数展示了Jensen类型不等式的一些新改进。这些改进已在离散版本和完整版本中得到了证明。上述结果使我们能够对Jensen不等式和Jensen–Steffensen不等式进行一些改进。同样，我们提出了反向Jensen和Jensen–Steffensen不等式的一些改进。然后，作为改进的詹森不等式的结果，我们为幂，几何和拟算术手段推导出了一些新的界线，还获得了埃尔米特-哈达玛德差距的界线和霍德尔不等式的改善。最后，作为改进的Jensen不等式的应用，我们为各种散度和Zipf–Mandelbrot熵提出了一些新的界线。