In this paper, we are interested in finding new polynomial inequalities by using the modified Smirnov operator. This operator carries a polynomial f(z) into <Equation ID="Equ31"> <EquationSource Format="TEX">$\begin{aligned} \tilde{\mathbb {S}}_a[f](z)=(1+az)f'(z)-naf(z), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mover accent="true"> <mi mathvariant="double-struck">S</mi> <mo stretchy="false">~</mo> </mover> <mi>a</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>n</mi> <mi>a</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where a is a complex number in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|z| \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The obtained results are compact generalization of several well-known polynomial inequalities.

Some compact generalizations of Bernstein-type inequalities preserved by modified Smirnov operator

期刊论文
发表日期：

2026年5月7日

查看全文

Deepak Kumar,
Dinesh Tripathi,
Sunil Hans

摘要

In this paper, we are interested in finding new polynomial inequalities by using the modified Smirnov operator. This operator carries a polynomial f(z) into $\begin{aligned} \tilde{\mathbb {S}}_a[f](z)=(1+az)f'(z)-naf(z), \end{aligned}$ $\begin{matrix} {\tilde{S}}_{a} [f] (z) = (1 + a z) f^{'} (z) - n a f (z), \end{matrix}$ where a is a complex number in $|z| \le 1$ $| z | \leq 1$ . The obtained results are compact generalization of several well-known polynomial inequalities.