发表机构
Karamanoğlu Mehmetbey University; Babeş-Bolyai University; Óbuda University（卡拉曼奥卢·梅赫梅特贝伊大学; 巴贝什-博尧伊大学; 奥比乌大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明并推广了Baricz关于第一类贝塞尔函数的Turán型不等式猜想，通过分析归一化Turán表达式的局部最小值性质，将结果推广至一般贝塞尔函数、库仑波函数等，并给出复平面包含性质。
AI 中文摘要
本文旨在证明Á. Baricz关于第一类贝塞尔函数的一个猜想，该猜想改进了O. Szász证明的第一类贝塞尔函数的经典Turán型不等式。其思路是考虑第一类贝塞尔函数的归一化Turán表达式，并证明在第一类贝塞尔函数的两个相邻零点之间，该归一化Turán表达式的分支具有局部最小值，且这些最小值构成一个严格递增的收敛序列，其各项关于阶数严格递减。此外，我们将此结果推广到一般贝塞尔函数和正则库仑波函数，并采用类似方法证明了纯虚阶修正贝塞尔函数和抛物柱面函数的锐利Turán型不等式。另外，我们证明了关于第一类贝塞尔函数逐次最小值严格递减性质的一个复解析类似物：利用H.S. Wilf意义上的从属因子序列技术，得到第一类贝塞尔函数Turán表达式在复平面中的包含性质。本文所采用的技术可能有助于处理涉及其他振荡特殊函数的Turán式或归一化Turán式的类似问题。
英文摘要
In this paper our aim is to prove a conjecture of Á. Baricz on Bessel functions of the first kind, which improves the classical Turán type inequality for Bessel functions of the first kind proved by O. Szász. The idea is to consider the normalized Turán expression for Bessel functions of the first kind and to show that between two consecutive zeros of the Bessel functions of the first kind the branch of this normalized Turán expression has a local minimum, and the minimum values form a strictly increasing convergent sequence, whose terms are strictly decreasing with respect to the order. Moreover, we extend this result to general Bessel functions and regular Coulomb wave functions, and we use a similar approach to show sharp Turán type inequalities for modified Bessel functions of purely imaginary order and parabolic cylinder functions. In addition, we prove a complex analogue of the result on Bessel functions of the first kind about the strictly decreasing property of the successive minimum values: an inclusion property in the complex plane of the Turán expression for Bessel functions of the first kind by using the subordinating factor sequence technique in the sense of H.S. Wilf. The techniques employed in the paper may be useful to treat similar problems where Turánians or normalized Turánians of other oscillatory special functions appear.
Comments30 pages, 3 figures
