<p>In the present paper, we introduce the subclasses <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{b}^{}\left( q,\phi \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mo>∑</mo> <mrow> <mi>b</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mfenced close=")" open="("> <mi>q</mi> <mo>,</mo> <mi>ϕ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{b}^{}\left( \alpha ,q,\phi \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mo>∑</mo> <mrow> <mi>b</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mfenced close=")" open="("> <mi>α</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>ϕ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of meromorphic functions <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$f\left( z\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq6.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="205" /> </InlineMediaObject> <EquationSource Format="TEX">$1+\frac{1}{b}\left[ -\frac{qzD_{q}^{}f(z)}{f(z)}-1\right] \prec \phi (z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> <mfenced close="]" open="["> <mo>-</mo> <mfrac> <mrow> <mi>q</mi> <mi>z</mi> <mmultiscripts> <mi>D</mi> <mrow> <mi>q</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> </mfenced> <mo>≺</mo> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq7.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="469" /> </InlineMediaObject> <EquationSource Format="TEX">$1+\frac{1}{b}\left[ \frac{-\left( 1-\frac{\alpha }{q}\right) qzD_{q}^{}f\left( z\right) +\alpha qzD_{q}^{}\left[ zD_{q}^{}f\left( z\right) \right] }{\left( 1-\frac{\alpha }{q}\right) f\left( z\right) -\alpha zD_{q}^{}f\left( z\right) }-1\right] \prec \phi (z)\ (b\in \mathbb {C} ^{}=\mathbb {C}\backslash \left\{ 0\right\} ,\ $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> <mfenced close="]" open="["> <mfrac> <mrow> <mo>-</mo> <mfenced close=")" open="("> <mn>1</mn> <mo>-</mo> <mfrac> <mi>α</mi> <mi>q</mi> </mfrac> </mfenced> <mi>q</mi> <mi>z</mi> <mmultiscripts> <mi>D</mi> <mrow> <mi>q</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>+</mo> <mi>α</mi> <mi>q</mi> <mi>z</mi> <mmultiscripts> <mi>D</mi> <mrow> <mi>q</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mfenced close="]" open="["> <mi>z</mi> <mmultiscripts> <mi>D</mi> <mrow> <mi>q</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mfenced> </mrow> <mrow> <mfenced close=")" open="("> <mn>1</mn> <mo>-</mo> <mfrac> <mi>α</mi> <mi>q</mi> </mfrac> </mfenced> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>-</mo> <mi>α</mi> <mi>z</mi> <mmultiscripts> <mi>D</mi> <mrow> <mi>q</mi> </mrow> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>f</mi> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> </mfenced> <mo>≺</mo> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="4pt" /> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∈</mo> <mmultiscripts> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>=</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="true">\</mo> </mrow> <mfenced close="}" open="{"> <mn>0</mn> </mfenced> <mo>,</mo> <mspace width="4pt" /> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="261" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \in \mathbb {C}\backslash (0,1],\ \operatorname {Re}(\alpha )\ge 0,\ 0<q<1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="true">\</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="4pt" /> <mo>Re</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mspace width="4pt" /> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, respectively. Sharp bounds for the Fekete-Szegö functional <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2024_1223_Article_IEq9.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$\left| a_{1}-\mu a_{0}^{2}\right| $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="|" open="|"> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>μ</mi> <msubsup> <mi>a</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> </mfenced> </math></EquationSource> </InlineEquation> are obtained.</p>

Fekete-szegö results for certain class of meromorphic functions using $q-$derivative operator

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2025年2月7日

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A. O. Mostafa,
G. M. El-Hawsh

摘要

In the present paper, we introduce the subclasses $\sum _{b}^{*}\left( q,\phi \right) $ $\sum_{b}^{*} (q, ϕ)$ and $\sum _{b}^{*}\left( \alpha ,q,\phi \right) $ $\sum_{b}^{*} (α, q, ϕ)$ of meromorphic functions $f\left( z\right) $ $f (z)$ satisfying $1+\frac{1}{b}\left[ -\frac{qzD_{q}^{*}f(z)}{f(z)}-1\right] \prec \phi (z)$ $1 + \frac{1}{b} [- \frac{q z D_{q}^{*} f (z)}{f (z)} - 1] ≺ ϕ (z)$ and $1+\frac{1}{b}\left[ \frac{-\left( 1-\frac{\alpha }{q}\right) qzD_{q}^{*}f\left( z\right) +\alpha qzD_{q}^{*}\left[ zD_{q}^{*}f\left( z\right) \right] }{\left( 1-\frac{\alpha }{q}\right) f\left( z\right) -\alpha zD_{q}^{*}f\left( z\right) }-1\right] \prec \phi (z)\ (b\in \mathbb {C} ^{*}=\mathbb {C}\backslash \left\{ 0\right\} ,\ $ $1 + \frac{1}{b} [\frac{- (1 - \frac{α}{q}) q z D_{q}^{*} f (z) + α q z D_{q}^{*} [z, D_{q}^{*}, f, (z)]}{(1 - \frac{α}{q}) f (z) - α z D_{q}^{*} f (z)} - 1] ≺ ϕ (z) (b \in C^{*} = C \ \{0\},$ $\alpha \in \mathbb {C}\backslash (0,1],\ \operatorname {Re}(\alpha )\ge 0,\ 0<q<1)$ $α \in C \ (0, 1], Re (α) \geq 0, 0 < q < 1)$ , respectively. Sharp bounds for the Fekete-Szegö functional $\left| a_{1}-\mu a_{0}^{2}\right| $ $|a_{1} - μ a_{0}^{2}|$ are obtained.

Fekete-szegö results for certain class of meromorphic functions using \(q-\)derivative operator

摘要