Let p (z) be a polynomial of degree n and for any real or complex number α, let Dαp (z)= np (z)+(... more Let p (z) be a polynomial of degree n and for any real or complex number α, let Dαp (z)= np (z)+(α− z) p′(z) denote the polar derivative of the polynomial p (z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros ...
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
This paper contains some results for algebraic polynomials in the complex plane involving the pol... more This paper contains some results for algebraic polynomials in the complex plane involving the polar derivative that are inspired by some classical results of Bernstein. Obtained results yield the polar derivative analogues of some inequalities giving estimates for the growth of derivative of lacunary polynomials.
Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspire... more Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspired by the classical Bernstein-type inequalities that relate the uniform norms of polynomials and their derivatives on the unit circle, are investigated. The obtained results sharpen as well as generalize some already known L δ-estimates between polynomials and their polar derivatives.
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
This paper deals with the problem of finding some upper bound estimates for the maximum modulus o... more This paper deals with the problem of finding some upper bound estimates for the maximum modulus of the derivative and higher order derivatives of a complex polynomial on a disk under the assumption that the polynomial has no zeros in another disk. The estimates obtained strengthen the well-known inequality of Ankeny and Rivlin about polynomials.
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
In this paper, we establish some inequalities for rational functions with prescribed poles and re... more In this paper, we establish some inequalities for rational functions with prescribed poles and restricted zeros in the sup-norm on the unit circle in the complex plane. Generalizations and refinements of rational function inequalities of Govil, Li, Mohapatra and Rodriguez are obtained.
This paper focuses on the problem concerning the location and the number of zeros of polynomials ... more This paper focuses on the problem concerning the location and the number of zeros of polynomials in a specific region when their coefficients are restricted with special conditions. We obtain extensions of some classical results concerning the number of zeros of polynomials in a prescribed region by imposing the restrictions on the moduli of the coefficients, the real parts(only) of the coefficients, and the real and imaginary parts of the coefficients.
Let p (z) be a polynomial of degree n and for any real or complex number α, let Dαp (z)= np (z)+(... more Let p (z) be a polynomial of degree n and for any real or complex number α, let Dαp (z)= np (z)+(α− z) p′(z) denote the polar derivative of the polynomial p (z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros ...
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
This paper contains some results for algebraic polynomials in the complex plane involving the pol... more This paper contains some results for algebraic polynomials in the complex plane involving the polar derivative that are inspired by some classical results of Bernstein. Obtained results yield the polar derivative analogues of some inequalities giving estimates for the growth of derivative of lacunary polynomials.
Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspire... more Some Zygmund-type integral inequalities for the polar derivatives of complex polynomials, inspired by the classical Bernstein-type inequalities that relate the uniform norms of polynomials and their derivatives on the unit circle, are investigated. The obtained results sharpen as well as generalize some already known L δ-estimates between polynomials and their polar derivatives.
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
This paper deals with the problem of finding some upper bound estimates for the maximum modulus o... more This paper deals with the problem of finding some upper bound estimates for the maximum modulus of the derivative and higher order derivatives of a complex polynomial on a disk under the assumption that the polynomial has no zeros in another disk. The estimates obtained strengthen the well-known inequality of Ankeny and Rivlin about polynomials.
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
In this paper, we establish some inequalities for rational functions with prescribed poles and re... more In this paper, we establish some inequalities for rational functions with prescribed poles and restricted zeros in the sup-norm on the unit circle in the complex plane. Generalizations and refinements of rational function inequalities of Govil, Li, Mohapatra and Rodriguez are obtained.
This paper focuses on the problem concerning the location and the number of zeros of polynomials ... more This paper focuses on the problem concerning the location and the number of zeros of polynomials in a specific region when their coefficients are restricted with special conditions. We obtain extensions of some classical results concerning the number of zeros of polynomials in a prescribed region by imposing the restrictions on the moduli of the coefficients, the real parts(only) of the coefficients, and the real and imaginary parts of the coefficients.
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Papers by Abdullah Mir