We consider functions of the type <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1926_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="207" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z)=z+a_2z^2+a_3z^3+\cdots $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>z</mi> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>a</mi> <mn>3</mn> </msub> <msup> <mi>z</mi> <mn>3</mn> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> </math></EquationSource> </InlineEquation> from a family of all analytic and univalent functions in the unit disk. Let F be the inverse function of f, given by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1926_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="179" /> </InlineMediaObject> <EquationSource Format="TEX">$F(w)=w+\sum _{n=2}^{\infty }A_nw^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>w</mi> <mo>+</mo> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>∞</mi> </msubsup> <msub> <mi>A</mi> <mi>n</mi> </msub> <msup> <mi>w</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> defined on some disk <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1926_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$|w|\le r_0(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>w</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we find the sharp bounds of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1926_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$\big | |A_{n+1}|-|A_n|\big |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">|</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">|</mo> </mrow> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1926_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1,\,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em" /> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, for some subclasses of univalent functions.

On the Difference of Initial Successive Coefficients of Inverse Functions

期刊论文
发表日期：

2025年7月14日

查看全文

Vasudeavarao Allu,
Vibhuti Arora

摘要

We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots $ $f (z) = z + a_{2} z^{2} + a_{3} z^{3} + \dots$ from a family of all analytic and univalent functions in the unit disk. Let F be the inverse function of f, given by $F(w)=w+\sum _{n=2}^{\infty }A_nw^n$ $F (w) = w + \sum_{n = 2}^{\infty} A_{n} w^{n}$ defined on some disk $|w|\le r_0(f)$ $| w | \leq r_{0} (f)$ . In this paper, we find the sharp bounds of $\big | |A_{n+1}|-|A_n|\big |$ $| | A_{n + 1} | - | A_{n} | |$ , for $n=1,\,2$ $n = 1, 2$ , for some subclasses of univalent functions.