In the study of various q-versions of the Bernstein polynomials, a significant attention is paid to their limit operators. The present work focuses on the impact of the limit q-Stancu operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_{\infty }^{q,\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>S</mi> <mrow> <mi>∞</mi> </mrow> <mrow> <mi>q</mi> <mo>,</mo> <mi>α</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> on the analytic properties of functions when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<q<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha >0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is shown that for every <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f\in C[0,1],$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$S_{\infty }^{q,\alpha }f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>∞</mi> </mrow> <mrow> <mi>q</mi> <mo>,</mo> <mi>α</mi> </mrow> </msubsup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> admits an analytic continuation into the disk <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{z: |z+\alpha /(1-q)|<1+\alpha /(1-q)\}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>z</mi> <mo>:</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo>+</mo> <mi>α</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In addition, it is proved that the more derivatives f has at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x=1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the wider this disk becomes. Further, if f is infinitely differentiable at <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$x=1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> then the function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$S_{\infty }^{q,\alpha }f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mrow> <mi>∞</mi> </mrow> <mrow> <mi>q</mi> <mo>,</mo> <mi>α</mi> </mrow> </msubsup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> is entire. Finally, some growth estimates for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$(S_{\infty }^{q,\alpha }f)(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>S</mi> <mrow> <mi>∞</mi> </mrow> <mrow> <mi>q</mi> <mo>,</mo> <mi>α</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are obtained.

How Analytic Properties of Functions Influence Their Images Under the Limit q-Stancu Operator?

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2026年1月10日

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Ovgu Gurel,
Sofiya Ostrovska,
Mehmet Turan

摘要

In the study of various q-versions of the Bernstein polynomials, a significant attention is paid to their limit operators. The present work focuses on the impact of the limit q-Stancu operator $S_{\infty }^{q,\alpha }$ $S_{\infty}^{q, α}$ on the analytic properties of functions when $0<q<1$ $0 < q < 1$ and $\alpha >0.$ $α > 0 .$ It is shown that for every $f\in C[0,1],$ $f \in C [0, 1],$ the function $S_{\infty }^{q,\alpha }f$ $S_{\infty}^{q, α} f$ admits an analytic continuation into the disk $\{z: |z+\alpha /(1-q)|<1+\alpha /(1-q)\}.$ ${z : | z + α / (1 - q) | < 1 + α / (1 - q)} .$ In addition, it is proved that the more derivatives f has at $x=1,$ $x = 1,$ the wider this disk becomes. Further, if f is infinitely differentiable at $x=1,$ $x = 1,$ then the function $S_{\infty }^{q,\alpha }f$ $S_{\infty}^{q, α} f$ is entire. Finally, some growth estimates for $(S_{\infty }^{q,\alpha }f)(z)$ $(S_{\infty}^{q, α} f) (z)$ are obtained.