Entire functions having small logarithmic sums over certain discrete subsets

It is known how to obtain a uniform estimate of e.g. a polynomial in terms of its logarithmic sum over the integers provided that the sum is sufficiently small. This result is generalized here and we obtain estimates in terms of logarithmic sums taken over certain discrete subsets of the real axis.     

Approximation by polynomials with small coefficients

Translated from Matematicheskie Zametki, Vol. 57, No. 1, pp. 150–153, January, 1995.     

Approximation by rational functions with polynomials of positive coefficients as the denominators

The present paper proves that if f(x) ∈ C[0,1], changes its sign exactly l times at 0 ＜ y1 ＜ y2 ＜y1＜1 in(0,1),then there exists a pn (x)пn( ),such that |f(x)-p(x)/pn(x)|≤ Cωφ(f,n-1/2),where ρ(x) is defined by ρ(x)=l∏i=1(x - yi), if f(x) ≥ 0 for x ∈ (yl, 1),-1∏i=1(x-yi), iff(x) ＜ 0 for x ∈ (y1,1).which improves and generalizes the result of [7].     

On a certain classes of meromorphic functions with positive coefficients

In this paper certain classes of meromorphic functions in punctured unit disk are defined. Some properties including coefficient inequalities, convolution and other results are investigated.     

On a subclass of certain k-starlike functions with negative coefficients

The aim of the present paper is to show some properties of functions belonging to the class (k, n, α)− . The obtained results extend the results by Silverman [3].     

Approximation of analytic functions by polynomials with integer coefficients

For some classes of analytic functions we estimate the rate of approximation by polynomials with integer coefficients.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 182–188, February, 1996.     

Uniqueness theorems of entire functions sharing a nonzero complex number with their difference operators

In this paper, we deal with a uniqueness question of entire functions sharing a nonzero complex number with their difference operators. The results in this paper improve Theorem 1.1 in Liu and Yang (Arch. Math. 92 (2009), 270–278) and deal with Question 1 in Liu and Yang (2009), where the entire functions are of finite orders.     

A conjecture regarding the logarithmic coefficients of univalent functions

One considers the class S of functions, regular and univalent in ¦Z¦<1 and normalized by the expansion f(z)=Z + C₂Z² +.... By the logarithmic coefficients of the function f (z) S one means the coefficients of the expansion Earlier, the author had formulated the following conjecture: for any function f(z) S, for each z (0,1) one has the inequality In this paper this conjecture is proved for spiral-shaped functions and for functions from S with real coefficients and under some additional assumptions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 135–143, 1983.     

A linearization for operator polynomials with coefficients in certain operator ideals

Summary For an operator polynomial with coefficients in certain Von Neumann-Schatten classes a linearization is constructed. The linearization also belongs to a Von Neumann-Schatten class, and its regularized Fredholm determinant determines the spectral properties of the original polynomial.     

Bands with a small number of ternary term functions

Bands with at most 9 essentially ternary term functions are characterized in this paper. May 5, 1999     

A sharp inequality for the logarithmic coefficients of univalent functions

We prove the sharp inequality

for the logarithmic coefficients of a normalized univalent function in the unit disk.

     

On the number of zeros of certain harmonic polynomials

Using techinques of complex dynamics we prove the conjecture of Sheil-Small and Wilmshurst that the harmonic polynomial , 1$">, has at most complex zeros.