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Generalizations of certain well-known inequalities for rational functions

On the Smirnov-Type Inequality for Polynomials

Smirnov and Bernstein-type inequalities, taking into account higher-order coefficients and free terms of polynomials

Since the time of Gauss there has been continued interest in those problems that center around the location of the zeros of a polynomial. Short expositions of the theory have been given, and a lengthier one by Dieudonné, but now, with the publication of this work, the third in the series of Mathematical Surveys of the American Mathematical Society, a comprehensive treatise on the subject has been made available. The lengthy, 20-page bibliography attests to the aim of the author to cover the material thoroughly; and considerable work must have been involved to integrate this heterogeneous material into such unity as is possible. Even so, the book could not have been kept to 161 pages had the author not adopted the plan of putting considerable theoretical material into the problems that appear at the end of most sections. Where necessary, hints are given with the problems, a wise gesture as readers of the book will see. In a well-written preface there is a brief account of the origin of the subject, and a statement of the two central problems that the book is to treat, together with a declaration of the principal working tools. The zeros of a polynomial f(z) are functions of the coefficients. Thus one problem is to specify regions, determined by these coefficients, in which the zeros lie. Again, with ƒ(z) one may associate a second polynomial (frequently this is the derivative ƒ' (s))> and thus arises the problem of relating the location of the zeros of the associate to the location of those oîf(z). There are also variants of these problems, of which something will be said as the individual chapters are discussed. As for the methods and tools that are used, they are classical. Thus, among other theorems, those of Cauchy, Rouché, and Hurwitz on zeros of analytic functions are of considerable use; and many theorems involve, either in proof or in statement, geometric and algebraic concepts of an elementary nature. Chapter I is introductory. The above-mentioned classical theorems are stated and proofs given. Then various interpretations, from physics, geometry, and function-theory, are given for the zeros of the rational function

The Geometry of the Zeros of a Polynomial in a complex variable. By M. Marden. Pp. ix, 183. $5. 1949. Mathematical Surveys, 3. (American Mathematical Society)

In this paper we consider a modified version of Smirnov operator and obtain some Bernstein-type inequalities preserved by this operator. In particular, we prove some results which in turn provide the compact generalizations of some well-known inequalities for polynomials.

Bernstein-type inequalities preserved by modified smirnov operator

The question raised in this article goes back to the problem posed by the famous chemist D. I. Mendeleev in 1887 (solved by A. A. Markov in 1889). In the next 100 years, the Mendeleev problem was repeatedly modificated and solved. Its essence is in the description of conditions under which the inequality ∣f(z)∣ ≤ ∣F(z)∣ for polynomials f and F and for z from a fixed set implies the inequality ∣L[f](z)∣ ≤ ∣L[F](z)∣ for some differential operator L. In the presented paper, we consider a differential operator of special type and arbitrary order. In particular, we obtain a sharp upper estimate for higher order derivatives of arbitrary polynomial in terms of the polynomial values.

Generalization of the Smirnov Operator and Differential Inequalities for Polynomials

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On the Smirnov-Type Inequality for Polynomials

Generalization of the Smirnov Operator and Differential Inequalities for Polynomials