Generalized Bessel functions of the first kind
This book deals with properties of generalized Bessel functions of order \(p\), which are defined as solutions of the differential equation \[ z^2w''(z)+ bz w'(z)+ [cz^2- p^2+(1- b)p]w(z)= 0, \] which in case \(b= c=1\) reduces to the (ordinary) Bessel differential equation. Moreover by specialization of the constants \(b\) and \(c\) one obtains the differential equation for the modified Bessel and for the spherical Bessel functions. So all these functions can be considered in a unified manner. In particular the function \[ w_p(z)= \sum^\infty_{n=0} {(-c)^n\over n!\Gamma[p+ n+ (b+ 1)/2]} \biggl({z\over 2}\biggr)^{2n+p},\quad z\in\mathbb{C} \] is called generalized Bessel function of the first kind of order \(p\). The author writes: ``In this book our aim is to study the generalized Bessel functions of the first kind from the point of view of the complex analysis on the one hand, and on the other hand from the point of view of the classical analysis.'' This leads on the one hand to results related to univalence, starlikeness, convexity and others and on the other hand to results on the extension of trigonometric inequalities like Mitrinovic, Matragan, Redheffer, Cusa, Jordan, Grünbaum, Landen, van der Corput and Askey, to generalized Bessel functions of the first kind. Chapter 1 deals with the transfer for some analytical properties of the (ordinary) Bessel functions of the first kind to their generalization. In Chapter 2 geometric properties of generalized Bessel functions as univalence, starlikeness and convexity properties are derived. The last Chapter 3 contains inequalities as indicated above, many taken from original papers of the author. In an Appendix, ``Conjectures'' and ``Open problems'', formulated by the author and co-authors of original papers are stated as \texttt{Matlab} programs for graphs. The References contain 246 items, 27 of them are papers of the author partly with co-authors. To my knowledge, the results are presented in a monograph for the first time.
scientific article; zbMATH DE number 5712790