A Generalization of the Krätzel Function and Its Applications

In this paper, we introduce new functions Yρ,rν(x) as a generalization of the Krätzel function. We investigate recurrence relations, Mellin transform, fractional derivatives, and integral of the function Yρ,rν(x). We show that the function Yρ,rν(x) is the solution of differential equations of fractional order

Meromorphic univalent function with negative coefficient

Let Mn be the classes of regular functions f(z)=z−1+a0+a1z+… defined in the annulus 0<|z|<1 and satisfying ReIn+1f(z)In+1f(z)>0, (n∈ℕ0), where I0f(z)=f(z), If(z)=(z−1−z(z−1)−2)∗f(z), Inf(z)=I(In−1f(z)), and ∗ is the Hadamard convolution. We denote by Γn=Mn⋃Γ, where Γ denotes the class of functions of the form f(z)=z−1+∑k=1∞|ak|zk. We obtained that relates the modulus of the coefficients to starlikeness for the classes Mn and Γn, and coefficient inequalities for the classes Γn

On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis in L-2(0, 1)