In this paper, we obtain some inequalities involving linear differential operator N for polynomials over complex domain and study the dependence of modulus of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_594_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="297" /> </InlineMediaObject> <EquationSource Format="TEX">$P(Rz)-\beta P(rz)+\delta \{ (\frac{R+1}{r+1})^n-|\beta |\}P(rz)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>β</mi> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">{</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mfrac> <mrow> <mi>R</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <mo>-</mo> <mrow> <mo stretchy="false">|</mo> <mi>β</mi> <mo stretchy="false">|</mo> <mo stretchy="false">}</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all complex numbers <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_594_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha , \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_594_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$ \delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_594_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="168" /> </InlineMediaObject> <EquationSource Format="TEX">$|\alpha |\ge 1, |\beta |\le 1, |\delta |\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>α</mi> <mo stretchy="false">|</mo> <mo>≥</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">|</mo> <mi>β</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">|</mo> <mi>δ</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> on the extreme values of |P(z)| over the boundary of unit circle after successive applications of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_594_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$D_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> and N. Our results besides yielding some interesting inequalities as special cases also generalize recently obtained inequalities related to the content.

Inequalities involving polar derivative and N-operator

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发表日期：

2025年5月14日

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F. A. Bhat

摘要

In this paper, we obtain some inequalities involving linear differential operator N for polynomials over complex domain and study the dependence of modulus of $P(Rz)-\beta P(rz)+\delta \{ (\frac{R+1}{r+1})^n-|\beta |\}P(rz)$ $P (R z) - β P (r z) + δ {{(\frac{R + 1}{r + 1})}^{n} - | β |} P (r z)$ for all complex numbers $\alpha , \beta $ $α, β$ and $ \delta $ $δ$ with $|\alpha |\ge 1, |\beta |\le 1, |\delta |\le 1$ $| α | \geq 1, | β | \leq 1, | δ | \leq 1$ on the extreme values of |P(z)| over the boundary of unit circle after successive applications of $D_{\alpha }$ $D_{α}$ and N. Our results besides yielding some interesting inequalities as special cases also generalize recently obtained inequalities related to the content.