Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m in T(M)$. In this paper we study the domination number of $T(Gamma(M))$ a...

Let R be a finite region of the s-plane and let L2(R) denote the class of functions f(z) each analytic in R with //b|/(8)| 2dS< °°. Let the points ßi, ß2, ß3, • • • be given interior to R and let 0„(z) be that function of class L2(R) for which n(ßi) =n(ß2) = • • • =n(ßn-i) =0, n(ßn) = 1, and which minimizes ffs\ <¡>n(z) \ 2dS over the class L2(R). If the points ßi, ß2, ß3, ■ ■ ■ are...

In this paper, we show that for any finite order entire function $f(z)$, the function of the form $f(z)^{n}[f(z+c)-f(z)]^{s}$ has no nonzero finite Picard exceptional value for all nonnegative integers $n, s$ satisfying $ngeq 3$, which can be viewed as a different result on Hayman conjecture. We also obtain some uniqueness theorems for difference polynomials of entire functions sharing one comm...

It is well-known that PSL(2, Z) is the group generated by the linear-fractional transformations x : z → − z and y : z → z−1 z , which satisfy the relations x = y = 1. We denote this modular group by G = 〈x, y : x = y = 1〉. Let n = km, where m is a square-free positive integer and k is any non zero integer. Then Q∗( √ n) = { √ n c : a, c and b = a −n c are integers and (a, b, c) = 1} is a G-subs...

For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...

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