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In this paper, we establish some new inequalities in the plane that are inspired by some classical Turán-type inequalities that relate the norm of a univariate complex coefficient polynomial and its derivative on the unit disk. The obtained results produce various inequalities in the integral-norm of a polynomial that are sharper than the previous ones while taking into account the placement of the zeros and some of the extremal coefficients of the underlying polynomial. [ABSTRACT FROM AUTHOR]

For a rational function r(z) having no zeros outside a unit disk, Li et al. (J Lond Math Soc 51:523–531, 1995) proved: | r ′ (z) | ≥ | B ′ (z) | | r (z) | , for | z | = 1 , where B (z) is the Blashke product. This result was improved and generalized in various ways and a recent generalization was obtained by Mir (Indian J Pure Appl Math 51:749–760, 2020). In this paper, we prove some results which include as special cases many Turán-type polynomial inequalities proved earlier. [ABSTRACT FROM AUTHOR]

The famous T. Suffridge polynomials have many extremal properties: the maximality of coefficients when the leading coefficient is maximal; the zeros of the derivative are located on the unit circle; the maximum radius of stretching the unit disk with the schlicht normalization F (0) = 0 , F ′ (0) = 1 ; the maximum size of the unit disk contraction in the direction of the real axis for univalent polynomials with the normalization F (0) = 0 , F (1) = 1. However, under the standard symmetrization method F (z T) T , these polynomials become functions which are not polynomials. How can we construct the polynomials with fold symmetry that have properties similar to those of the Suffridge polynomial? What values will the corresponding extremal quantities take in the above-mentioned extremal problems? The paper is devoted to solving these questions for the case of the trinomials F (z) = z + a z 1 + T + b z 1 + 2 T . Also, there are suggested hypotheses for the general case in the work. [ABSTRACT FROM AUTHOR]

In this paper, we prove some inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. We further obtain the polar derivative generalizations of the obtained results. The obtained results produce various inequalities that are sharper than the previous ones while taking into account the placement (absolute value) of the zeros and the extremal coefficients of the polynomial. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. [ABSTRACT FROM AUTHOR]