Some inequalities for power series with applications

Ibrahim, Alawiah; Dragomir, Sever S; Darus, Maslina

doi:10.1080/10652469.2012.693489

Cited by 8 publications

(9 citation statements)

References 14 publications

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“…For other similar results and applications for special functions see the research papers [1], [4]- [6] and the survey [7].…”

Section: Introductionmentioning

confidence: 91%

New Inequalities of CBS-Type for Power Series of Complex Numbers

Dragomir

2016

Fasciculi Mathematici

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Let $f(\lambda ) = \sum\nolimits_{n = 0}^\infty {a_n \lambda ^n } $ be a function defined by power series with complex coefficients and convergent on the open disk D(0, R) ⊂ ℂ, R > 0. In this paper we show amongst other that, if α, z ∈ ℂ are such that |α|, |α| |z|2 < R, then $$\left| {f(\alpha )f(\alpha z^2 ) - f^2 (\alpha z)} \right| \le f_A \left( {\left| \alpha \right|} \right) f_A \left( {\left| \alpha \right|\left| z \right|^2 } \right) - \left| {f_A (\left| \alpha \right|z)} \right|^2 .$$ where $f_A (z) = \sum\nolimits_{n = 0}^\infty {\left| {\alpha _n } \right|z^n } $ . Applications for some fundamental functions defined by power series are also provided.

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“…For other similar results and applications for special functions see the research papers [1], [4]- [6] and the survey [7].…”

Section: Introductionmentioning

confidence: 91%

New Inequalities of CBS-Type for Power Series of Complex Numbers

Dragomir

2016

Fasciculi Mathematici

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“…For inequalities of power series as complex functions, see [101,102,103] and the references therein. A(a, b), the geometric mean G (a, b), the logarithmic mean L(a, b), the identric mean I(a, b), the root square mean Q(a, b), and the power mean M r (a, b) of order r, defined, respectively, by…”

Section: Inequalities For Power Seriesmentioning

confidence: 99%

Topics in Special Functions III

Anderson

Вуоринен

Zhang

2014

Analytic Number Theory, Approximation Theory, and Special Functions

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The study of quasiconformal maps led the first two authors in their joint work with M. K. Vamanamurthy to formulate open problems or questions involving special functions [14,16]. During the past two decades, many authors have contributed to the solution of these problems. However, most of the problems posed in [14] are still open.The present paper is the third in a series of surveys by the first two authors, the previous papers [19,21] being written jointly with the late M. K. Vamanamurthy. The aim of this series of surveys is to review the results motivated by the problems in [14,16] and related developments during the past two decades. In the first of these we studied classical special functions, and in the next we focused on special functions occurring in the distortion theory of quasiconformal maps. Regretfully, Vamanamurthy passed away in 2009, and the remaining authors dedicate the present work as a tribute to his memory. For an update to the bibliographies of [19] and [21] the reader is referred to [23].In 1993 the following monotone rule was derived [17, Lemma 2.2]. Though simple to state and easy to prove by means of the Cauchy Mean Value Theorem, this l'Hôpital Monotone Rule (LMR) has had wide application to special functions by many authors. Vamanamurthy was especially skillful in the application of this rule. We here quote the rule as it was restated in [20, Theorem 2].

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“…For various other properties of Gamma function and its related functions, refer to [1,6,7,9,19,20,48,49,70,81,84,88,90,94,98,99,101,102,122,123]. Barnes [11] defined the double Gamma function 2 D 1=G satisfying each of the following properties:…”

Section: Gamma Functionmentioning

confidence: 99%

“…For inequalities of power series as complex functions, see [99][100][101] and the references therein.…”

Section: Theorem 65 ([105]mentioning

confidence: 99%