It is known that if f is holomorphic in the open unit disc (mathbb{D}) of the complex plane and if, for some c > 0, ∣f(z)∣ ⩽ 1/(1−∣z∣2)c, (z in mathbb{D}), then ∣f′(z)∣ ⩽ 2(c+1)/(1−∣z∣2)c+1. We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in D. In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.
Given a graph G = (V, E), if we can partition the vertex set V into two nonempty subsets V1 and V2 which satisfy Δ(G[V1]) ⩽ d1 and Δ(G[V2]) ⩽ d2, then we say G has a (({{rm{Delta }}_{{d_1}}},,{{rm{Delta }}_{{d_2}}}))-partition. And we say G admits an (({F_{d_{1}}}, {F_{d_{2}}}))-partition if G[V1] and G[V2] are both forests whose maximum degree is at most d1 and d2, respectively. We show that every planar graph with girth at least 5 has an (F4, F4)-partition.
Let G be a finite group and construct a graph Δ(G) by taking G {1} as the vertex set of Δ(G) and by drawing an edge between two vertices x and y if 〈x, y〉 is cyclic. Let K(G) be the set consisting of the universal vertices of Δ(G) along the identity element. For a solvable group G, we present a necessary and sufficient condition for K(G) to be nontrivial. We also develop a connection between Δ(G) and K(G) when ∣G∣ is divisible by two distinct primes and the diameter of Δ(G) is 2.
Let j ⩾ 2 be a given integer. Let Hk* be the set of all normalized primitive holomorphic cusp forms of even integral weight k ⩾ 2 for the full modulo group SL(2, ℤ). For f ∈ Hk*, denote by ({{rm{lambda }}_{{rm{sy}}{{rm{m}}^j}{kern 1pt} f}}(n)) the nth normalized Fourier coefficient of jth symmetric power L-function (L(s, symjf)) attached to f. We are interested in the average behaviour of the sum
$$sumlimits_{scriptstyle n, = ,a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x atop scriptstyle ,,,,,,,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{rm{)}} in ,{{mathbb{Z}}^6}} {{rm{lambda }}_{{rm{sy}}{{rm{m}}^j},fleft( n right),}^2}$$where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：