SOME GENERALIZATIONS FOR OPIAL'S INEQUALITY INVOLVING SEVERAL FUNCTIONS AND THEIR DERIVATIVES OF ARBITRARY ORDER ON ARBITRARY TIME SCALES

KARPUZ B., ÖZKAN U. M.

MATHEMATICAL INEQUALITIES & APPLICATIONS, cilt.14, sa.1, ss.79-92, 2011 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 14 Sayı: 1
  • Basım Tarihi: 2011
  • Doi Numarası: 10.7153/mia-14-07
  • Dergi Adı: MATHEMATICAL INEQUALITIES & APPLICATIONS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.79-92
  • Anahtar Kelimeler: Muirhead's inequality, Opial's inequality, Taylor's formula, time scales, weights
  • Dokuz Eylül Üniversitesi Adresli: Hayır

Özet

In this paper, some various types of Opial's inequality involving several functions and their higher-order derivatives are presented on arbitrary time scales. The well-known Muirhead's inequality is employed to obtain very interesting results. While dealing with higher-order derivatives, the generalized Taylor's formula and the generalized polynomials are used to simplify our proofs too. Our new results generalize and extend the existing results in the literature, and some of the works done by Pachpatte. Moreover, our results are not only new for arbitrary time scales, but also new for the continuous and the discrete cases.