Geodesic
Some Hermite–Hadamard type inequalities for the square norm in Hilbert spaces
Annales Universitatis Mariae Curie-Skłodowska. Mathematica, Tome 75 (2021) no. 2, pp. 31-44 Cet article a éte moissonné depuis la source Library of Science

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Let ( H;〈· ,·〉) be a complex Hilbert space and f:[0,∞ )→ℝ be convex (concave) on [0,∞ ). If x, y∈ H with Re 〈 x,y〉≥ 0, then f( ‖ x‖ ^2+Re 〈 x,y〉 +‖ y‖ ^2/3) ≤( ≥) ∫_0^1f( ‖( 1-t) x+ty‖^2) dt ≤( ≥) 1/3[ f( ‖ x‖^2) +f[ Re 〈 x,y〉] +f(‖ y‖ ^2) ] . Some examples for power functions and exponential are also provided.
Keywords: Convex functions, Hermite–Hadamard inequality, midpoint inequality, power and exponential functions 
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Dragomir, Silvestru Sever. Some Hermite–Hadamard type inequalities for the square norm in Hilbert spaces. Annales Universitatis Mariae Curie-Skłodowska. Mathematica, Tome 75 (2021) no. 2, pp. 31-44. http://geodesic.mathdoc.fr/item/AUM_2021_75_2_a2/

[1] Barnett, N. S., Cerone, P., Dragomir, S. S., Some new inequalities for Hermite-Hadamard divergence in information theory, in: Stochastic Analysis and Applications, Vol. 3, Nova Sci. Publ., Hauppauge, NY, 2003, 7–19. Preprint RGMIA Res. Rep. Coll. 5 (2002), Art. 8, 11 pp. [Online https://rgmia.org/papers/v5n4/NIHHDIT.pdf]

[2] Cerone, P., Dragomir, S. S., Mathematical Inequalities. A Perspective, CRC Press, Boca Raton, FL, 2011.

[3] Dragomir, S. S., An inequality improving the first Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2) (2002), Art. 31, 8 pp.

[4] Dragomir, S. S., An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (3) (2002), Art. 35, 8 pp.

[5] Dragomir, S. S., Operator Inequalities of Ostrowski and Trapezoidal Type, Springer Briefs in Mathematics. Springer, New York, 2012.

[6] Dragomir, S. S., Operator Inequalities of the Jensen, Cebysev and Gruss Type, Springer Briefs in Mathematics. Springer, New York, 2012.

[7] Dragomir, S. S., Pearce, C. E. M., Selected Topics on Hermite–Hadamard Inequalities and Applications, RGMIA Monographs, 2000. [Online http://rgmia.org/monographs/hermite hadamard.html]

[8] Pecaric, J., Dragomir, S. S., A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat. (Sarajevo) 7 (1991), 103–107.

[9] Pecaric, J. E., Proschan, F., Tong, Y. L., Convex Functions, Partial Orderings and Statistical Applications, Academic Press Inc., Boston, MA, 1992.