<p>For <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1963_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\lambda , \ \alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>λ</mi> <mo>,</mo> <mspace width="4pt" /> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1963_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {U(\alpha ,\lambda )}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a sub-class of non-Bazilevič functions defined by <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1963_Article_IEq5.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="198" /> </InlineMediaObject> <EquationSource Format="TEX">$\left| f^{\prime }(z)\left( z/f(z)\right) ^{\alpha +1}-1\right| <\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="|" open="|"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mfenced close=")" open="("> <mi>z</mi> <mo stretchy="false">/</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mfenced> <mrow> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mn>1</mn> </mfenced> <mo><</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation>. In this article, we present the bounds on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1963_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$||a_{3}|-|a_{2}||$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for Taylor’s coefficients for the functions in the class <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1963_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {U(\alpha ,\lambda )}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">U</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also establish the same bounds for the inverse, logarithmic and logarithmic inverse coefficients. All bounds presented in this paper are sharp.</p>

On the Difference of Coefficients of Class $\mathcal {U(\alpha ,\lambda )}$

期刊论文
发表日期：

2025年9月3日

查看全文

Umar Raza,
Mohsan Raza

摘要

For $0<\lambda , \ \alpha <1$ $0 < λ, α < 1$ , let $\mathcal {U(\alpha ,\lambda )}$ $U (α, λ)$ be a sub-class of non-Bazilevič functions defined by $\left| f^{\prime }(z)\left( z/f(z)\right) ^{\alpha +1}-1\right| <\lambda $ $|f^{'} (z) {(z / f (z))}^{α + 1} - 1| < λ$ . In this article, we present the bounds on $||a_{3}|-|a_{2}||$ $| | a_{3} | - | a_{2} | |$ for Taylor’s coefficients for the functions in the class $\mathcal {U(\alpha ,\lambda )}$ $U (α, λ)$ . We also establish the same bounds for the inverse, logarithmic and logarithmic inverse coefficients. All bounds presented in this paper are sharp.

On the Difference of Coefficients of Class \(\mathcal {U(\alpha ,\lambda )}\)

摘要