Let $\psi : \mathbb {N} \to [0,1/2]$ be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi (q)/q$ , provided series $\sum _{q=1}^\infty \varphi (q) / q$ is divergent. In present paper, we establish a quantitative version...

Let _(x, !)r |!| p&2 ! be a p-Laplacian type operator and consider the Hodge decomposition _(x, Du)=D.+H, div H=0. A standard elliptic theory asserts that &D.&q (p&1) C &Du& p&1 q for all q>p&1. There has been considerable recent interest in the validity of the reverse estimate &Du& p&1 q C &D.&q (p&1) for q>p&1 in the regularity study of certain geometrical mappings. In this paper, we give a r...

It is suggested that the (p, q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions 2 (1994) 15-36) can be considered as a special case of a more general (P, Q)-hypergeometric series. 1. Introduction I propose a general (P, Q)-hypergeometric series. In this, I am inspired mainly by the paper of Burban and Klimyk [1] titled " P, Q-differentiation, P, ...