In this work, a new inequality by sharpening the well-known Holder inequality by means of a theorem based on abstract convexity is derived. | Turk J Math (2016) 40: 438 – 444 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article The sharpening H¨ older inequality via abstract convexity G¨ ultekin TINAZTEPE∗ Vocational School of Technical Sciences, Akdeniz University, Antalya, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this work, a new inequality by sharpening the well-known H¨ older inequality by means of a theorem based on abstract convexity is derived. Key words: Abstract convexity, functional inequalities, H¨ older inequality, global optimization 1. Introduction The applications of abstract convexity in different areas are known (see [2, 3, 4, 5, 6, 7, 8, 9]). One of them is the application to inequality theory. For instance, for different function classes, Hermite–Hadamard type inequalities were derived by different authors in [2, 3, 4, 8]. Another application of abstract convexity to inequality theory is to sharpen known inequalities in [7] and [1]. In [1], the sharper versions for well-known inequalities among the generalized arithmetic, geometric, and harmonic means is given by using abstract convexity, and it is shown that the presented sharping scheme does not derive a sharper inequality for every inequality satisfying related conditions, such as, for example, Cauchy–Schwarz and Minkowski inequalities. In this paper, the H¨older inequality is studied and investigated in the frame of abstract convexity in light of [1]. Sharper inequality for the H¨older inequality is derived, and also by using this result, we present sharper inequality for the Cauchy–Schwarz inequality. The structure of the paper is as follows: in the second section, certain concepts of abstract convexity, an important theorem to be applied to optimization theory and the H¨older inequality, are given. In the third section, the H¨older inequality is considered, results are presented as .
