Differential subordination of a harmonic mean to a linear function
The much studied subject of subordination of a function \(f\), analytic in the unit disc, to a function \(g\), univalent in the unit disc, defined via a composition of \(g\) with a normalised member of the class \(B\) of bounded analytic functions, has been extended to the concept of differential subordination [\textit{S. S. Miller} and \textit{P. T. Mocanu}, Differential subordinations: theory and applications. New York, NY: Marcel Dekker (2000; Zbl 0954.34003)]. Miller and Mocanu [loc. cit.] discuss the concept of differential subordination as a complex analogue of the concept of differential inequality aboundantly used in real analysis. The prime examples of the functions for which a differential inequality provides information about the range of the function in the complex case are furnished by the class \(P\) of analytic functions with positive real part as well as convex, star-like and close-to-convex classes. The focus of the paper under review is differential subordination to a linear dominant, plus some study of selected convex dominants. The particular brand of differential subordination studied is related to the harmonic mean. For two complex numbers \(a\) and \(b\), and a number \(t\) from the interval \([0,1]\), the harmonic mean of \(a\) and \(b\) is the ratio \(ab/[b+t(a-b)]\), provided the denominator is not zero. For any nonconstant function \(p\), defined on the unit disc, and a fixed \(t\) from the interval \((0,1)\), the families of functions of the type \[ [p(z)p(z)+zp'(z)]/[p(z)+(1-t)zp'(z)] \] which are either analytic in the unit disc or possess only removable singularities there and analytic extentions to the unit disc, are studied in the context of subordination to a linear function. The authors of the paper under review work with a technical lemma from [loc. cit.] and use it to prove a theorem that is a generalization of their own result from [Bull. Soc. Sci. Lett. Łódź, Sér. Rech. Déform. 64, No. 2, 29--40 (2014; Zbl 1320.30046)]. In the process, they manage to generalize a case of the first order Euler differential subordination for a nonlinear function in the book of Miller and Mocanu [loc. cit.], and obtain a special case of a subordination result from [\textit{D. J. Hallenbeck} and \textit{S. Ruscheweyh}, Proc. Am. Math. Soc. 52, 191--195 (1975; Zbl 0311.30010)].
scientific article; zbMATH DE number 6958788
scientific article