In this paper, we consider the generalized Marcum function of the second kind as an analogous function of the so-called generalized Marcum Q -function. We provide the log-convexity (log-concavity) property for its unit complement and improve some of our previous results on it. One of the transformed functions of the generalized Marcum function of the second kind is discussed in details in this paper. This form of the generalized Marcum function of the second kind supplies various important inequalities. We also discuss the Turán type inequality for the generalized Marcum Q -function. Additionally, we provide the bounds for the generalized Marcum function of the second kind as well as for its symmetric difference.

中文翻译：

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第二类广义 Marcum 函数的函数不等式和界限

在本文中，我们将第二类广义 Marcum 函数视为所谓广义 Marcum Q 函数的类比函数。我们为其单元补集提供了对数凸性（log-concavity）属性，并改进了我们之前的一些结果。本文详细讨论了第二类广义 Marcum 函数的变换函数之一。第二类广义 Marcum 函数的这种形式提供了各种重要的不等式。我们还讨论了广义 Marcum Q 函数的 Turán 型不等式。此外，我们提供了第二类广义 Marcum 函数及其对称差的边界。