Geodesic
On Willett's, Godunova--Levin's, and Rozanova's Opial-Type Inequalities with Related Stolarsky-Type Means
Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 803-819.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider generalizations of Opial's inequality due to Willett, Godunova, Levin, and Rozanova. Cauchy-type mean-value theorems are proved and used in studying Stolarsky-type means defined by the obtained inequalities. Also, a method of producing $n$-exponentially convex and exponentially convex functions is applied.
Keywords: Willett's inequality, Godunova–Levin's inequality, Rozanova's inequality, Cauchy mean-value theorems, exponential convexity, Stolarsky means. 
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M. Andric; A. Barbir; J. Pečarić. On Willett's, Godunova--Levin's, and Rozanova's Opial-Type Inequalities with Related Stolarsky-Type Means. Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 803-819. http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a0/

[1] Z. Opial, “Sur une inégalité”, Ann. Polon. Math., 8 (1960), 29–32 | MR | Zbl

[2] R. P. Agarwal, P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Math. Appl., 320, Kluwer Acad. Publ., Dordrecht, 1995 | MR | Zbl

[3] D. Willett, “The existence-uniqueness theorem for an $n$-th order linear ordinary differential equation”, Amer. Math. Monthly, 75:2 (1968), 174–178 | DOI | MR | Zbl

[4] E. K. Godunova, V. I. Levin, “Ob odnom neravenstve Maroni”, Matem. zametki, 2:2 (1967), 221–224 | MR | Zbl

[5] G. I. Rozanova, “Integralnye neravenstva s proizvodnymi i proizvolnymi vypuklymi funktsiyami”, Uchenye zapiski Mosk. gos. ped. in-ta im. V. I. Lenina, 460 (1972), 58–65 | MR

[6] M. Anwar, J. Jakšetić, J. Pečarić, Atiq Ur Rehman, “Exponential convexity, positive semi-definite matrices and fundamental inequalities”, J. Math. Inequal., 4:2 (2010), 171–189 | DOI | MR | Zbl

[7] J. Pečarić, J. Perić, “Improvements of the Giaccardi and Petrović inequality and related Stolarsky type means”, An. Univ. Craiova Ser. Mat. Inform., 39:1 (2012), 65–75 | MR | Zbl

[8] J. E. Pečarić, F. Proschan, Y. C. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Math. Sci. Engrg., 187, Acad. Press, Boston, MA, 1992 | MR | Zbl

[9] J. Jakšetić, J. Pečarić, “Exponential convexity method”, J. Convex Anal., 20:1 (2013), 181–197 | MR | Zbl

[10] D. V. Widder, The Laplace Transform, Princeton Math. Ser., 6, Princeton Univ. Press, Princeton, NJ, 1941 | MR | Zbl