Inspired by the advent of bi-unique range sets [2], we obtain a new bi-unique range sets, with smallest cardinalities ever for the derivatives of meromorphic functions which improves all the results obtained so far in some sense including a result of Banerjee-Bhattacharjee [4]. Furthermore at the last section we pose an open question for future research.

In 1932 R. P. Agnew present a definition for Deferred Cesàro mean. Using this definition R. P. Agnew present inclusion theorems for the deferred and none Deferred Cesàro means. This paper is part 2 of a series of papers that present extensions to the notion of double Deferred Cesàro means. Similar to part 1 this paper uses this definition and the notion of regularity for four dimensional matrices, to present extensions and variations of the inclusion theorems presented by R. P. Agnew in [2].

In this paper we have examined various spectrum of the operator $D\left(p,q,r,s\right)$ on the sequence space $\chi^{2}$ defined by Musielak Orlicz function.

The relationship between algebraic soliton metrics and self-similar solutions of geometric evolution equations on Lie groups is investigated. After discussing the general relationship between algebraic soliton metrics and self-similar solutions to geometric evolution equations, we investigate the cross curvature flow and the second order renormalization group flow on simply-connected, three-dimensional, unimodular Lie groups, providing a complete classification of left invariant algebraic solitons that give rise to self-similar solutions of the corresponding flows on such spaces.

In this paper, with the aid of weighted value sharing we study the uniqueness problems of meromorphic functions when certain nonlinear differential polynomials generated by them share a nonconstant polynomial with weight two. The result of the paper not only improves the results due to the present first author [Bull. Math. Anal. Appl., 2(2010), 106-118] and of Zhang and Xu [Comput. Math. Appl., 61(2011), 722-730], at the same time finds a possible answer of an open question posed by Zhang and Xu.

In this note we generalize the fact that in topological vector spaces the algebraic sum of closed set $A$ and compact set $B$ is closed. We also prove some conditions that are equivalent to reflexivity of Banach spaces.

In 1955, Waadeland considered the class of m-fold symmetric starlike functions of the form $f_{m}(z)=z+\sum_{n=1}^{\infty }a_{mn+1}z^{mn+1}$; $m\geq 1$; $|z|\lt1$ and obtained the sharp coefficient bounds $|a_{mn+1}|\leq\left[ (2/m+n-1)!\right] /\left[ (n!)(2/m-1)!\right] $. Pommerenke in 1962, proved the same coefficient bounds for m-fold symmetric close-to-convex functions. Nine years later, Keogh and Miller confirmed the same bounds for the class of m-fold symmetric Bazilevic functions. Here we will show that these bounds can be improved even further for the m-fold symmetric bi-close-to-convex functions. Moreover, our results improve those corresponding coefficient bounds given by Srivastava et al that appeared in 7(2) (2014) issue of this journal. A function is said to be bi-close-to-convex in a simply connected domain if both the function and its inverse map are close-to-convex there.

In this paper, we acquire the general solution of the undecic functional equation $$\begin{align*} & f(x+6y)-11f(x+5y)+55f(x+4y)-165f(x+3y)+330f(x+2y)\\ & \qquad\qquad -462f(x+y)+462f(x)-330f(x-y)+165f(x-2y)-55f(x-3y)\\ & \quad\qquad\qquad\qquad\qquad\qquad +11f(x-4y)-f(x-5y)=39916800f(y). \end{align*}$$ We also obtain the generalized Ulam-Hyers stability of the above functional equation in quasi-$\beta$-normed spaces using fixed point method. Moreover, we investigate the pertinent stabilities of the above functional equation using sum of powers of norms, product of powers of norms and mixed product-sum of powers of norms as upper bounds. We also present a counter-example for non-stability of the above functional equation in singular case.

In this paper, we introduce Matlis flat modules as a generalization of copure flat modules and give their characterizations. We prove that if $R$ is a commutative Artinian ring and $S \subset R$ is a multiplicative set, then $S^{-1}M$ is a Matlis flat $S^{-1}R$-module for any Matlis flat $R$-module $M$. Also we prove that every module has Matlis flat preenvelope over commutative Artinian rings.

In this paper, we consider the stationary fractional reaction-diffusion equations with Riemann-Liouville boundary conditions $$\begin{aligned} &{_{x}}D_{T}^{\alpha}({_{0}}D_{x}^{\alpha}u(x)) + {_{0}}D_{x}^{\beta}({_{x}}D_{T}^{\beta}u(x)) = f(x,u(x)),\;\;x\in (0,T),\\ &\lim_{x\to 0} {_{0}}I_{x}^{1-\alpha}u(x) = \lim_{x\to T} {_{x}}I_{T}^{1-\beta}u(x) = 0. \end{aligned}$$

(0.1)

where $0\lt \alpha , \beta \lt 1$ and $f\in C([0,T] \times \mathbb{R}, \mathbb{R})$. Under suitable conditions on the nonlinearity $f$, we study the multiplicity of weak solutions of (0.1) by using the genus in the critical point theory.

Gabor frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Finding general and verifiable conditions which imply that the Gabor systems are Gabor frames is among the core problems in time-frequency analysis. In this paper, we give some simple and sufficient conditions that ensure a Gabor system $\left\{M_{u(m)b}T_{u(n)a}g=:\chi_{m}(bx)g\big(x-u(n)a\big)\right\}_{ m,n\in\mathbb N_0}$ to be a frame for $L^2(K)$. The conditions proposed are stated in terms of the Fourier transforms of the Gabor system's generating functions.

In the paper, the authors present the Schur-convexity of the absolute of the logarithm of the Catalan–Qi function and prove the logarithmically complete monotonicity of the Catalan–Qi function.

By applying hypergeometric operator on Hilbert space, the author introduces a new class of meromorphic multivalent functions with an arbitrary fixed point omega. Properties such as coefficient inequalities, distortion bounds and extreme points were derived. Furthermore, the effect of this operator on functions in this class was also investigated.