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On geometric properties of the generating function for the Ramanujan sequence

RAMANUJAN JOURNAL 2018年第1期46卷 173-188页

作者： Bakan, Andrew Ruscheweyh, Stephan Salinas, Luis Natl Acad Sci Ukraine Inst Math UA-01601 Kiev Ukraine Univ Wurzburg Inst Math D-97074 Wurzburg Germany UTFSM Dept Informat Valparaiso Chile

The Ramanujan sequence , defined as , , , has been studied on many occasions and in many different contexts. Adell and Jodra (Ramanujan J 16:1-5, 2008) and Koumandos (Ramanujan J 30:447-459, 2013) showed, respectively, that the sequences and are completely monotone. In the present paper, we establish that the sequence is also completely monotone. Furthermore, we prove that the analytic function is universally starlike for every in the slit domain . This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by Ruscheweyh et al. (Israel J Math 171:285-304, 2009), namely that the function is universally convex.

关键词： Ramanujan's sequence Completely monotone sequences universally starlike functions universally convex functions

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UNIVERSAL convexITY AND UNIVERSAL STARLIKENESS OF POLYLOGARITHMS

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2015年第2期143卷 717-729页

作者： Bakan, Andrew Ruscheweyh, Stephan Salinas, Luis Natl Acad Sci Ukraine Inst Math UA-01601 Kiev Ukraine Univ Wurzburg Inst Math D-97074 Wurzburg Germany UTFSM Dept Informat Valparaiso Chile

A deep result of J. Lewis (1983) shows that the polylogarithms Li-alpha(z) := Sigma(infinity)(k = 1) z(k)/k(alpha) map the open unit disk D centered at the origin one-to-one onto convex domains for all alpha >= 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain Lambda := C \ [1,infinity), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Li-alpha(z) maps an arbitrary open disk or half-plane in. one-to-one onto a convex domain for every alpha >= 1.

关键词： convex functions polylogarithms universally convex functions universally starlike functions Pick functions

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universally prestarlike functions as convolution multipliers

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MATHEMATISCHE ZEITSCHRIFT 2009年第3期263卷 607-617页

作者： Ruscheweyh, Stephan Salinas, Luis Univ Wurzburg Math Inst D-97074 Wurzburg Germany Univ Tecn F Santa Maria Dept Informat Valparaiso Chile

universally prestarlike functions (of order alpha <= 1) in the slit domain Lambda := C\[1, infinity] have recently been introduced in Ruscheweyh et al. (Israel J Math, to appear). This notation generalizes the corresponding one for functions in the unit disk D (and other circular domains in C). In this paper we study the behaviour of universally prestarlike functions under the Hadamard product. In particular it is shown that these function classes (with alpha fixed), are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.

关键词： universally prestarlike functions universally convex functions Convolution invariance Bernardi-Libera transform

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