<p>For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {\overline{B}}(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi mathvariant="script">B</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the class of non-Bazilevič functions defined by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Re\left( f^{\prime }(z)\left( z/f(z)\right) ^{\alpha +1}\right) >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>e</mi> <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> </mfenced> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this article, we present the bounds of <InlineEquation ID="IEq4"> <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> and <InlineEquation ID="IEq5"> <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 the class <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {\overline{B}}(\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi mathvariant="script">B</mi> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also establish the similar results for the logarithmic and logarithmic inverse coefficients for the same class. All bounds presented in this paper are sharp.</p>

On the Difference of Coefficients of Non-Bazilevič Functions

期刊论文
发表日期：

2025年10月29日

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Rashid Ali,
Mohsan Raza

摘要

For $0<\alpha <1$ $0 < α < 1$ , let $\mathcal {\overline{B}}(\alpha )$ $\bar{B} (α)$ be the class of non-Bazilevič functions defined by $Re\left( f^{\prime }(z)\left( z/f(z)\right) ^{\alpha +1}\right) >0$ $R e (f^{'}, (z), {(z / f (z))}^{α + 1}) > 0$ . In this article, we present the bounds of $||a_{3}|-|a_{2}||$ $| | a_{3} | - | a_{2} | |$ and $||A_{3}|-|A_{2}||$ $| | A_{3} | - | A_{2} | |$ for the class $\mathcal {\overline{B}}(\alpha )$ $\bar{B} (α)$ . We also establish the similar results for the logarithmic and logarithmic inverse coefficients for the same class. All bounds presented in this paper are sharp.