Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> be the class of functions f(z) of the form <Equation ID="Equ19"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_Equ19.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="443" /> </MediaObject> <EquationSource Format="TEX">$ f(z)=z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots , (p\in \mathbb {N}=\{1,2,3,\ldots \}) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>z</mi> <mi>p</mi> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow> <mi>p</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>z</mi> <mrow> <mi>p</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>that are analytic in the open unit disc <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="158" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {U}=\big \{ z\in \mathbb {C}: |z| <1\big \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">U</mi> <mo>=</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">{</mo> </mrow> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z)\in \mathcal {A}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="script">A</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, Nunokawa considered some conditions such that f(z) is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$p-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>valent in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {U}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">U</mi> </math></EquationSource> </InlineEquation>. Applying the results by Nunokawa, we discuss some interesting properties for functions <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1264_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z)\in \mathcal {A}_{p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="script">A</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Also, we give some examples for our results.

New sufficient conditions for p-valent functions

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2025年2月17日

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Hatun Özlem Güney,
Sevtap Sümer,
Shigeyoshi Owa

摘要

Let $\mathcal {A}_{p}$ $A_{p}$ be the class of functions f(z) of the form $ f(z)=z^{p}+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+\cdots , (p\in \mathbb {N}=\{1,2,3,\ldots \}) $ $f (z) = z^{p} + a_{p + 1} z^{p + 1} + a_{p + 2} z^{p + 2} + \dots, (p \in N = {1, 2, 3, \dots})$ that are analytic in the open unit disc $\mathbb {U}=\big \{ z\in \mathbb {C}: |z| <1\big \}$ $U = {z \in C : | z | < 1}$ . For $f(z)\in \mathcal {A}_{p}$ $f (z) \in A_{p}$ , Nunokawa considered some conditions such that f(z) is $p-$ $p -$ valent in $\mathbb {U}$ $U$ . Applying the results by Nunokawa, we discuss some interesting properties for functions $f(z)\in \mathcal {A}_{p}$ $f (z) \in A_{p}$ . Also, we give some examples for our results.