Gamma-Bazilevič Functions

For <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#947;</mi> <mo>&#8805;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display...

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Bibliographic Details
Main Authors: Sa’adatul Fitri, Derek K. Thomas
Format: Article
Language:English
Published: MDPI AG 2020-02-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/8/2/175
Description
Summary:For <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#947;</mi> <mo>&#8805;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>&#945;</mi> <mo>&#8805;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, we introduce the class <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi mathvariant="script">B</mi> <mn>1</mn> <mi>&#947;</mi> </msubsup> <mrow> <mo>(</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of Gamma&#8722;Bazilevič functions defined for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>&#8712;</mo> <mi mathvariant="double-struck">D</mi> </mrow> </semantics> </math> </inline-formula> by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>R</mi> <mi>e</mi> <mfenced separators="" open="{" close="}"> <msup> <mfenced separators="" open="[" close="]"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>&#8242;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>f</mi> <msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>&#8722;</mo> <mi>&#945;</mi> </mrow> </msup> <msup> <mi>z</mi> <mi>&#945;</mi> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mrow> <mo>&#8243;</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mi>f</mi> <mo>&#8242;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mo>+</mo> <mrow> <mo>(</mo> <mi>&#945;</mi> <mo>&#8722;</mo> <mn>1</mn> <mo>)</mo> </mrow> <mfenced separators="" open="(" close=")"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>&#8242;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mo>&#8722;</mo> <mn>1</mn> </mfenced> </mfenced> <mi>&#947;</mi> </msup> <msup> <mfenced open="[" close="]"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>&#8242;</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>f</mi> <msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>&#8722;</mo> <mi>&#945;</mi> </mrow> </msup> <msup> <mi>z</mi> <mi>&#945;</mi> </msup> </mrow> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>&#8722;</mo> <mi>&#947;</mi> </mrow> </msup> </mfenced> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. We shown that <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi mathvariant="script">B</mi> <mn>1</mn> <mi>&#947;</mi> </msubsup> <mrow> <mo>(</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is a subset of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, the class of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> Bazilevič functions, and is therefore univalent in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">D</mi> </semantics> </math> </inline-formula>. Various coefficient problems for functions in <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi mathvariant="script">B</mi> <mn>1</mn> <mi>&#947;</mi> </msubsup> <mrow> <mo>(</mo> <mi>&#945;</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are also given.
ISSN:2227-7390