Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Degot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit disk, the disk |z − a| ≤ 1 contains a critical point of P(z). Based on this result, we derive an explicit formula N(a) for each a ∈ (0, 1) and, consequently obtain a uniform bound N for all a ∈ [α, β] where 0 < α < β < 1. This (partially) addresses the questions posed in Degot’s paper.

In this paper we study analogues of the perfect splines for weighted Sobolev classes of functions defined on the half-line. Maximally oscillating splines play important role in the solution of certain extremal problems. In particular, using these splines, we characterize the modulus of continuity of the differential operator.

We provide sufficient conditions on the wavelets on a local field K for the wavelet system to form an unconditional basis for the Hardy space H1(K) and the Lebesgue spaces Lp(K), 1 < p < ∞.

This paper is a follow up of a recent work of the author and Y. Sawano [6], in 2019. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3], 1992. When the local integrability index s falls 1 with weights and t exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7] 2018. Later in the paper [6] we considered the case t = 1. This paper handles the remaining case 0 < t < 1.

The paper addresses questions on existence and structure of universal functions for spaces L1(E), E ⊂ [0, 1)2, with respect to the doubleWalsh system in the sense of signs of Fourier coefficients. It is shown that for each e > 0 one can find a measurable set Ee ⊂ [0, 1)2 with measure |Ee| > 1−e, such that by a proper modification of any integrable function f ∈ L1[0, 1)2 outside Ee one can get an integrable function f ∈ L1[0, 1)2 which is universal for L1(Ee) with respect to the double Walsh system in the sense of signs of Fourier coefficients.

In this paper we show that the maximum of a polynomial P ∈ ℂ[z] of degree d ≥ 2 in the unit disc can be bounded below by the sum of moduli of its two coefficients, say, for zs and zt under certain assumption on the pair s < t. We also show that this assumption on s, t cannot be removed or weakened and give several examples showing when this lower bound is (or is not) attained.

We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces we prove that some classical linear operators such as Calderon–Zygmund, convolution and Riesz potential operators are bounded.

In this paper, the calculus of functions with values in L-spaces is developed. We then consider nonlinear integral equations of Fredholm and Volterra types for functions with values in L-spaces. Such class of equations includes set-valued integral equations, fuzzy integral equations, and many others. We prove theorems of existence and uniqueness of the solutions of such equations and investigate data dependence of their solutions.

We give an estimation for the eigenvalues of matrix power functions. In particular, it has been shown that $$\lambda({(A+B)^p})\leq\lambda({2^{p-1}}({A^p}+{B^p}-\gamma I))\;(P \geq2)$$ for all positive semi-definite matrices A, B, where γ is a positive constant. This provides a sharper bound for the known estimation for eigenvalues.

We introduce polynomial sets of (p, q)-Appell type and give some characterizations of them. The algebraic properties of the set of all polynomial sequences of (p, q)-Appell type are studied. Next, we give a recurrence relation and a (p, q)-difference equation for those polynomials. Finally, some examples of polynomial sequences of (p, q)-Appell type, particularly, a set of (p, q)-Hermite polynomials are given and their three-term recurrence relation and a second order homogeneous (p, q)-difference equation are provided.