<p>The purpose of this paper is to find initial coefficient bounds <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|a_2|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$|a_3|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and Fekete-Szegö estimates for the functions that belong to a newly defined subclass <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathscr {G}\mathscr {B}^{\nu }_{\kappa }(l)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <msubsup> <mi mathvariant="script">B</mi> <mi>κ</mi> <mi>ν</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> consisting of analytic functions normalized by the conditions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f(0)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f^{\prime }(0)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> defined by the subordination to Limaçon-shaped domain. Similar results have been derived for the inverse function <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\log \dfrac{f(z)}{z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>log</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mi>z</mi> </mfrac> </mstyle> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\dfrac{z}{f(z)}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>z</mi> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </math></EquationSource> </InlineEquation>. Furthermore, applications of our results to certain distributions are defined and discussed using Hadamard product. Our findings generalize existing results and also introduce novel subclasses of univalent functions.</p>

Initial Estimates of Bazilevič Type Functions Defined by Limaçon-Shaped Domain

期刊论文
发表日期：

2025年9月3日

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Nanjundan Magesh,
Halit Orhan,
Dasanur Shivanna Raju,
Nagamangala Sathyananda Tejas

摘要

The purpose of this paper is to find initial coefficient bounds $|a_2|$ $| a_{2} |$ and $|a_3|$ $| a_{3} |$ and Fekete-Szegö estimates for the functions that belong to a newly defined subclass $\mathscr {G}\mathscr {B}^{\nu }_{\kappa }(l)$ $G B_{κ}^{ν} (l)$ consisting of analytic functions normalized by the conditions $f(0)=0$ $f (0) = 0$ and $f^{\prime }(0)=1$ $f^{'} (0) = 1$ defined by the subordination to Limaçon-shaped domain. Similar results have been derived for the inverse function $f^{-1}$ $f^{- 1}$ , $\log \dfrac{f(z)}{z}$ $log \frac{f (z)}{z}$ and $\dfrac{z}{f(z)}$ $\frac{z}{f (z)}$ . Furthermore, applications of our results to certain distributions are defined and discussed using Hadamard product. Our findings generalize existing results and also introduce novel subclasses of univalent functions.