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Название RU

Авторы RU

Аннотация RU
<p>Введены новые классы функций, обобщающие хорошо известные классы <em>p</em>-валентных функций, введенных ранее Т. Умезавой. Введены новые классы <em>p</em>-валентных функций, возникающих при применении формальных дифференциальных операторов и установлены достаточные условия принадлежности к ним. Также найдены определенные соотношения между этими классами.</p>

Ключевые слова RU

Литература RU

Название EN

Авторы EN

Аннотация EN
<p>The theory of analytic functions and more specific <i>p</i>-valent functions, is one of the most fascinating topics in one complex variable. There are many remarkable theorems dealing with extremal problems for a class of <i>p</i>-valent functions on the unit disk U. Recently, many researchers have shown great interests in the study of differential operator. The objective of this paper is to define a new generalized derivative operator of <i>p</i>-valent analytic functions of fractional power in the open unit disk U denoted by D^{<i>m</i>,<i>b</i>}<i>_{λ₁λ₂ <i>p</i>,α}f</i>(<i>z</i>). This operator generalized some well-known operators studied earlier, we mention some of them in the present paper. Motivated by the generalized derivative operator D^{<i>m</i>,<i>b</i>}_{λ₁λ₂ <i>p</i>,α}<i>f</i>(<i>z</i>) we introduce and investigate two new subclasses <i>S</i>^<i>m,b</i>_{λ1,λ2,<i>p</i>,α}(μ,ν) and <i>T</i><i>S</i>^<i>m,b</i>_{λ1,λ2,<i>p</i>,α}(μ,ν), which are subclasses of starlike <i>p</i>-valent analytic functions of fractional power with positive coefficients and starlike <i>p</i>-valent analytic functions of fractional power with negative coefficients, respectively. In addition, a sufficient condition for functions <i>f</i> ∈ Σ_<i>p</i>,α to be in the class <i>S</i>^<i>m,b</i>_{λ1,λ2,<i>p</i>,α}(μ,ν) and a necessary and sufficient condition for functions <i>f</i> ∈ <i>T</i>_<i>p</i>,α will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class <i>TS</i>^<i>m,b</i>_{λ1,λ2,<i>p</i>,α}(μ,ν).</p>

Ключевые слова EN

Литература EN
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