‎in this paper‎, ‎we investigate some results on natural metrics on the $mu$-normal functions and meromorphic $q_p$-classes‎. ‎also‎, ‎these classes are shown to be complete metric spaces with respect to the corresponding metrics‎. ‎moreover‎, ‎compact composition operators $c_phi$ and lipschitz continuous operators acting from $mu$-normal functions to the meromorphic $q_p$-classes are characte...

The quantum Rabi model (QRM) is widely recognized as an important in systems, particularly optics. Hamiltonian $H_{\text{Rabi}}$ known to have a parity decomposition $H_{\text{Rabi}} = H_{+} \oplus H_{-}$. In this paper, we give the explicit formulas for propagator of Schr\"odinger equation (integral kernel time evolution operator) and $H_{\pm}$ by Wick rotation (meromorphic continuation) corre...

For a constant k ∈ [0,∞) a normalized function f , analytic in the unit disk, is said to be k-uniformly convex if Re (1 + zf ′′(z)/f ′(z)) > k|zf ′′(z)/f ′(z)| at any point in the unit disk. The class of k-uniformly convex functions is denoted k-UCV (cf. [4]). The function g is said to be k-starlike if g(z) = zf ′(z) and f ∈ k-UCV. For analytic functions f, g, where f(z) = z + a2z + · · · and g...

where I is a finite set of distinct tuples (α0,α1, . . . ,αk) for which eachαi is a nonnegative integer, and the aᾱ are meromorphic functions in D = {z | |z| < 1}. For some index sets I, we determine conditions on aᾱ, whereby a meromorphic solution f of (1.1) in D will have finite order of growth as measured by the Ahlfors-Shimizu characteristic function. In [1], Bank investigated (1.1) where I...

The Hadwiger number H(S) of a topological disk S in R 2 is the maximal number of pairwise nonoverlapping translates of S that touch S. A conjecture of A. Bezdek., K. and W. Kuperberg [2] states that this number is at most eight for any starlike disk. A. Bezdek [1] proved that the Hadwiger number of a starlike disk is at most seventy five. In this note, we prove that the Hadwiger number of any c...

1. In addition to the study devoted to the general class of normalized regular univalent functions in the unit circle considerable attention has been given to certain subclasses of these functions such as starlike functions, convex functions, and others. In view of the important role played by symmetrization methods in recent work it seems worth while to study a further subclass—the class of ci...

Nevanlinna theory [Ne] was created to give a quantitative measure of the value distribution for meromorphic functions, for instance to measure the extent to which they approximate a finite number of points. We view a meromorphic function as a holomorphic map ƒ : C —• P into the projective line. The theory has various higher dimensional analogues, of which we shall later consider maps ƒ : C —• X...

In this paper, we derive conditions on the parameters a, b, c so that the function zF (a, b; c; z) is starlike in D, where F (a, b; c; z) denotes the classical hypergeometric function. We give some consequences of our results including some mapping properties of convolution operator.

Here we study the normalized Laplacian characteristics polynomial (L-polynomial) for trees and specifically for starlike trees. We describe how the L-polynomial of a tree depends on some topological indices. For which, we also define the higher order general Randić indices for matching and which are different from higher order connectivity indices. Finally we provide the multiplicity of the eig...