Gamma-Bazilevič Functions
Sa’adatul Fitri, Derek K. Thomas
Mathematics · 2020
doi:10.3390/math8020175 (opens in a new tab)
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Abstract
For <inline-formula> <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>≥</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>γ</mi> </msubsup> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of Gamma−Bazilevič functions defined for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>∈</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>′</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>−</mo> <mi>α</mi> </mrow> </msup> <msup> <mi>z</mi> <mi>α</mi> </msup> </mrow> </mfrac> </mstyle> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mrow> <mo>″</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mstyle> <mo>+</mo> <mrow> <mo>(</mo> <mi>α</mi> <mo>−</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>′</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>−</mo> <mn>1</mn> </mfenced> </mfenced> <mi>γ</mi> </msup> <msup> <mfenced open="[" close="]"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>z</mi> <msup> <mi>f</mi> <mo>′</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>−</mo> <mi>α</mi> </mrow> </msup> <msup> <mi>z</mi> <mi>α</mi> </msup> </mrow> </mfrac> </mstyle> </mfenced> <mrow> <mn>1</mn> <mo>−</mo> <mi>γ</mi> </mrow> </msup> </mfenced> <mo>></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>γ</mi> </msubsup> <mrow> <mo>(</mo> <mi>α</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>α</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>α</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>γ</mi> </msubsup> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are also given.
Abstract licence: CC BY
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