Bulletin of the Malaysian Mathematical Sciences Society 

Abstract: For a graph $$G = (V(G), E(G))$$ , let d(u), d(v) be the degrees of the vertices u, v in G. The first and second Zagreb indices of G are defined as $$ M_1(G) = \sum _{u \in V(G)} d(u)^2$$ and $$ M_2(G) = \sum _{uv \in E(G)} d(u)d(v)$$ , respectively. The first (generalized) and second Multiplicative Zagreb indices of G are defined as $$\Pi _{1,c}(G) = \prod _{v \in V(G)}d(v)^c$$ and $$\Pi _2(G) = \Pi _{uv \in E(G)} d(u)d(v)$$ , respectively. The (Multiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s. Denote by $${\mathcal {G}}_{n}$$ the set of all Eulerian graphs of order n. In this paper, we characterize Eulerian graphs with first three smallest and largest Zagreb indices and Multiplicative Zagreb indices in $${\mathcal {G}}_{n}$$ .

Abstract: Let $$\mathcal {S}^*_{SG}=\{f\in \mathcal {A}:zf'(z)/f(z)\prec 2/(1+e^{-z})\}$$. For this class, several radius estimates and coefficient bounds are obtained as well as structural formula, growth theorem, distortion theorem and inclusion relations are established. Further, let p be an analytic function such that $$p(0)=1$$. Sharp bounds on $$\beta \in \mathbb {R}$$ are determined for various first-order differential subordinations such as $$1+\beta zp'(z)/p^k(z)$$, $$p(z)+\beta zp'(z)/p^k(z)\prec 2/(1+e^{-z})$$ to imply that $$p(z)\prec (1+Az)/(1+Bz)$$, where $$-1\le B