Generalization of certain well-known polynomial inequalities

If is a polynomial of degree n having no zeros in |z|<1, then for |β|?1, it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45-51] that

     

On the zeros of certain composite polynomials and an operator preserving inequalities

The Ramanujan Journal - If all the zeros of nth degree polynomials f(z) and $$g(z) = \sum _{k=0}^{n}\lambda _k\left( {\begin{array}{c}n\\ k\end{array}}\right) z^k$$ respectively lie in the cricular...     

Differential inequalities for algebraic polynomials

We sharpen and supplement the results by V. I. Smirnov, A. Aziz, and Q. M. Dawood for algebraic polynomials which generalize the classical Bernstein and Erdos-Lax inequalities.     

On some inequalities for polynomials

We prove the equivalence between analogs of the Paley and Nikol’skii inequalities for any orthonormal system of functions and for almost periodic polynomials with arbitrary spectrum. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1289–1292, September, 1998.     

Integral inequalities for self-reciprocal polynomials

Let n ≥ 1 be an integer and let P _n be the class of polynomials P of degree at most n satisfying z ⁿ P(1/z) = P(z) for all z ∈ C. Moreover, let r be an integer with 1 ≤ r ≤ n. Then we have for all P ∈ P _n :

Generalization of Chebyshev polynomials

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 217–218, January–February, 1990.     

Generalization of hermite polynomials

A generalization of the Hermite polynomials is given. Some relations which let one construct orthogonal Hermite polynomials successively are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1524–1528, November, 1990.     

Generalization of Bernoulli polynomials

The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.     

Some Bernstein-type inequalities for trigonometric polynomials

The refinements of some well-known Bemstein-type inequalities for trigonometric polynomials are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 428–430, March, 1993.     

Generalization of the triangle and Ptolemy inequalities

The Euclidean triangle inequality generalizes to an alternating inequality for any oddsided polygon that can be inscribed in a circle; there is equality in the even cases. A generalization of Ptolemy's theorem follows by inversion. The results have Minkowskian analogues.     

Integral inequalities for algebraic and trigonometric polynomials

Doklady Mathematics - In this paper, we consider sharp estimates of integral functionals $int_0^{2pi } {phi (L|Lf_n (t)|)dt} $ for functions φ defined on the semiaxis (0, ∞) and...     

Integral inequalities for algebraic polynomials on the unit circle

Translated from Matematicheskie Zametki, Vol. 48, No. 4, pp. 7–18, October, 1990.     

Some inequalities for algebraic polynomials with the Laguerre weight

There is a series of publications which have considered inequalities of Markov–Bernstein–Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159–176; B.D. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (1982) 181–190; P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov–Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1 (1972) 15–25; A. Guessab, G.V. Milovanovic, Weighted L²-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244–249; I.I. Ibragimov, Some inequalities for algebraic polynomials, in: V.I. Smirnov (Ed.), Fizmatgiz, 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570–572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) (1957) 32–35; M.K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1944) 165–209; P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960) 373–378]). In this paper we find an inequality of the same type for algebraic polynomials on (0,∞) with the Laguerre weight function e^-xx^α (α>-1).     

New conjectured inequalities for zeros of Jacobi polynomials

Inequalities recently conjectured for all zeros of Jacobi polynomials \(P_n^{(\alpha,\beta)}\) of all degrees n are modified and conjectured to hold (in reverse direction) in considerably larger domains of the (α,β)-plane.