Transactions of NAS of Azerbaijan, Issue Mathematics, 37 (4), 102–110 (2017). Series of Physical-Technical and Mathematical Sciences On some Hardy-Sobolev’s type variable exponent inequality and its application Farman I. Mamedov · Sayali M. Mammadli · Yusuf Zeren Received: 05.05.2017 / Revised: 09.10.2017/ Accepted: 29.11.2017 Abstract. In this paper, it has been proved a Sobolev’s type variable exponent inequality    u(x) x(l - x)    p(x);(0,l) ≤ C l   u ′ (x)   p(x);(0,l) , ∀u ∈ ˙ W 1 p(.) (0,l) where the exponent function p : (0,l) → (1, ∞), is a monotone increasing near little neighborhood of origin and monotone decreasing near l satisfying the conditions:  l a t − 1 p ′ (t) dt t ≤ C 2 a − 1 p ′ (a) , and  l a t − 1 p ′ (l−t) dt t ≤ C 1 a − 1 p ′ (l−a) , for 0 < a < l. Applying this inequality and Browder-Minty theory methods, it has been proved an existence result of solution for some variable exponent equation. Keywords. variable exponent spaces, inequality, solvability, Dirichlet problem Mathematics Subject Classification (2010): 26D10, 42B37, 34L30 1 Introduction One of the results of this paper is the following assertion on variable exponent boundedness of the conjugate Hardy operator  l x f (t) dt on finite interval (0,l). F.I. Mamedov Mathematics and Mechanics Institute of National Academy of Sciences, Baku, Azerbaijan Oil and Gas Scientific Research Project Inst., SOCAR, Baku, Azerbaijan E-mail: farman-m@mail.ru S.M. Mammadli Mathematics and Mechanics Institute of National Academy of Sciences, Baku, Azerbaijan E-mail: sayka-426@mail.ru Y.Zeren Yildiz Technical University, Istanbul, Turkey E-mail: yusufzeren@hotmail.com