Non-normality, topological transitivity and expanding families

We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties.

An Analytical and Numerical Detour for the Riemann Hypothesis

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

A normal criterion concerning zero numbers

Let $$n \ge 4$$ n ≥ 4 be a positive integer, $$\mathcal {F}$$ F be a family of meromorphic functions in D and let $$a(z)(\not \equiv 0), b(z)$$ a ( z ) ( ≢ 0 ) , b ( z ) be two holomorphic functions in D. If, for any function $$f \in \mathcal { F}$$ f ∈ F , (1)$$f(z) \ne \infty $$ f ( z ) ≠ ∞ when $$a(z)=0$$ a ( z ) = 0 , (2) $$f'(z)-a(z)f^{n}(z)-b(z)$$ f ′ ( z ) - a ( z ) f n ( z ) - b ( z ) has at most one zero in D, then $$\mathcal {F}$$ F is normal in D.

On the Fekete–Szegö Problem for Meromorphic Functions Associated with p,q-Wright Type Hypergeometric Function

Making use of a post-quantum derivative operator, we define two classes of meromorphic analytic functions. For the considered family of functions, we aim to investigate the sharp bounds’ values in the case of the Fekete–Szegö problem. The study of the well-known Fekete–Szegö functional in the post-quantum calculus case for meromorphic functions provides new outcomes for research in the field. With the extended p,q-operator, we establish certain inequalities’ relations concerning meromorphic functions. In the final part of the paper, a new p,q-analogue of the q-Wright type hypergeometric function is introduced. This function generalizes the classical and symmetrical Gauss hypergeometric function. All the obtained results are sharp.

Characteristic estimation of differential polynomials

In this paper, we give the characteristic estimation of a meromorphic function f with the differential polynomials $f^{l}(f^{(k)})^{n}$ f l ( f ( k ) ) n and obtain that $$\begin{aligned} T(r,f)\leq M\overline{N} \biggl(r,\frac{1}{f^{l}(f^{(k)})^{n}-a} \biggr)+S(r,f) \end{aligned}$$ T ( r , f ) ≤ M N ‾ ( r , 1 f l ( f ( k ) ) n − a ) + S ( r , f ) holds for $M=\min \{\frac{1}{l-2},6\}$ M = min { 1 l − 2 , 6 } , integers $l(\geq 2)$ l ( ≥ 2 ) , $n(\geq 1)$ n ( ≥ 1 ) , $k(\geq 1)$ k ( ≥ 1 ) , and a non-zero constant a. This quantitative estimate is an interesting and complete extension of earlier results. The value distribution of a differential monomial of meromorphic functions is also investigated.