Journal of Prime Research in Mathematics

Abstract: The aim of the present paper is to investigate a new subclass of starlike functions of complex order \(b\neq 0\). Let \(f(z)=z+a_{2}z^{2}+…\) be an analytic function in the unit disc \(D=\{z| |z|<1\}\) which satisfies \(1+\frac{1}{b}(z\frac{f'(z)}{f(z)}-1)=\frac{1+A\omega z}{1+B\omega z}\), for some \(\omega \in \Omega\) and for all \(z \in D\). Then f is called a Janowski starlike function of complex order b, where A and B are complex numbers such that \(Re(1-A\overline{B})\geq |A-B|, im(1-A\overline{B}<|A-B|, |B|<1\) and \(\omega(z)\) ) is a Schwarz function in the unit disc D [1], [10], [12]. The class of these functions is denoted by \(S^{∗}(A, B, b)\). In this paper we will give the representation theorem, distortion theorem, two point distortion theorem, Koebe domain under the montel normalization, and coefficient inequality for this class.

Abstract: Let G be a graph with vertex set \(V = V (G)\) and edge set \(E = E(G)\) and let \(e = |E(G)|\) and \(v = |V (G)|\). A one-to-one map \(λ\) from \(V ∪ E\) onto the integers \(\{1, 2, …, v + e\}\) is called vertex magic total labeling if there is a constant \(k\) so that for every vertex \(x\), \(λ+\sum λ(xy)=k\). where the sum is over all vertices \(y\) adjacent to \(x\). Let us call the sum of labels at vertex x the weight \(w_{λ}(x)\) of the vertex under labeling \(λ\); we require \(w_{λ}(x) = k\) for all \(x\). The constant \(k\) is called the magic constant for \(λ\). In this paper, we present the vertex magic total labelings of disconnected graph, in particular, two copies of isomorphic generalized Petersen graphs \(2P(n, m)\), disjoint union of two non-isomorphic suns \(S_{m} ∪ S_{n}\) and t copies of isomorphic suns \(tS_{n}\).

Abstract: In this paper, we derive an upper bound for the size Ramsey number for a path \(P_3\) versus a friendship graph \(C_{3}^{t}\). Furthermore, some minimal Ramsey graph for a combination \((P_{3}, C_{3}^{t}) \)is presented. We also give an upper bound of the size Ramsey number for \(P_3\) versus \(P_n\).

Abstract: In this paper, we establish weak and strong convergence theorems of the three-step iterative sequences with errors for non-self nonexpansive mappings in uniformly convex Banach spaces. Our results extend and improve the recent ones announced by Naseer Shahzad and some others.