On some compositions of Hadamard type in classes of analytic functions

Analytic representations in the three-dimensional Frobenius problem

“…The ⊗-and •-products appear naturally in functional analysis, for example, in the Bieberbach conjecture for univalent analytic functions [23] and the Polya-Schoenberg conjecture for analytic convex mappings [24].…”

Valence properties of the solution of a differential equation

“…Epstein and Schoenberg [2] exhibited 30 DOUGLAS M. CAMPBELL AND V. SINGH a starlike univalent polynomial whose composition with a nonelementary univalent function was not even locally univalent (but was at most three valent). Finally, Loewner and Netanyahu [7] exhibited two close-to-convex functions whose Hadamard composition is not even locally univalent (but again they were unable to determine if H* contains functions of infinite valence). Loewner and Netanyahu's counterexample is to be contrasted with proof of the Polya-Schoenberg conjecture that f*g is starlike if / and g are starlike.…”

Open problems on univalent and multivalent functions

“…Here Robertson [54] has proved that if ffi = 3i(l), then 3C* = 3i(l). It was suggested by several mathematicians that the set *Üi(l) of univalent functions might be closed under the operation (10.3), but this conjecture was killed simultaneously by B. Epstein and I. J* Schoenberg [12], C. Loewner and E. Netanyahu [41 ], and W. K. Hayman [29]. This naturally raises the question of determining the maximum valence for functions in the class 3C*, and the radius of ^-valence for the class 3C*.…”