Motivated by the $k$-digamma function, we introduce a $k$-extension of the Nielsen's $beta$-function, and further study some properties and inequalities of the new function.

Let $$n\ne 0$$ be an integer. A set of m distinct positive integers $$\{a_1,a_2,\ldots ,a_m\}$$ is called a D(n)-m-tuple if $$a_ia_j + n$$ perfect square for all $$1\le i < j \le m$$ . k In this paper, we prove that $$\{k,k+1,c,d\}$$ $$D(-k)$$ -quadruple with $$c>1$$ , then $$d=1$$ The proof relies not only on standard methods in field (Baker’s linear forms logarithms and the hypergeometric met...

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...

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...

Definition 1. An element α of an extension field K of k is algebraic over k if it satisfies some monic polynomial with coefficients in k. Otherwise α is transcendental over k. A field extension K/k is algebraic if every α ∈ K is algebraic over k. A field k is algebraically closed if there is no proper algebraic extension K/k; that is, in any extension field K of k, any element α that is algebra...