Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals in the class , where is a fixed function and f is quasiconvex and fulfills a growth condition of the type

L⁻¹|z|^p(x)?f(x,ξ,z)?L(1+|z|^p(x)),     

Certain series attached to an even number of elliptic modular forms

For j=1,…,n let f_j(z) and g_j(z) be holomorphic modular forms for such that f_j(z)g_j(z) is a cusp form. We define a series

     

On the existence of universal series by trigonometric system

In this paper we prove the following: let ω(t) be a continuous function, increasing in [0,∞) and ω(+0)=0. Then there exists a series of the form

     

Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L²(R₊,S) by the differential expression

     

On multiply-connected Fatou components in iteration of meromorphic functions

Let be a transcendental meromorphic function with at most finitely many poles. We mainly investigated the existence of the Baker wandering domains of f(z) and proved, among others, that if f(z) has a Baker wandering domain U, then for all sufficiently large n, fn(U) contains a round annulus whose module tends to infinity as n→∞ and so for some 0<d<1,

     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place ,

H(φ)(x):=h(x,φ(x)),     

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was:

Theorem. (See Borwein, Choi, Yazdani, 2001.) Letf(z)=±z±z²±?±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer then

     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.     

On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities

Let B(H) be the C^*-algebra of all bounded linear operators acting on a complex Hilbert space H. In this note, we shall show that if S is an invertible normal operator in B(H) the following estimation holds

     

Lipschitz and piecewise-C regularity for scalar minimizers of affine simple integrals

Lipschitz, piecewise-C¹ and piecewise affine regularity is proved for AC minimizers of the “affine” integral , under general hypotheses on , , and with superlinear growth at infinity.The hypotheses assumed to obtain Lipschitz continuity of minimizers are unusual: ρ(·) and ?(·) are lsc and may be both locally unbounded (e.g., not in L_loc¹), provided their quotient ?/ρ(·) is locally bounded. As to h(·), it is assumed lsc and may take +∞ values freely.     

Inequalities for a polynomial and its derivative

Let , 1?μ?n, be a polynomial of degree n such that p(z)≠0 in |z|<k, k>0, then for 0<r?R?k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved

     

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

Let U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f(z)=z+a₂z²+? satisfying the condition