In this paper, we establish some new integral inequalities of Hermite–Hadamard type for s-convex functions by using the Holder–Iscan integral inequality. We also compare our new results with the known results and show that the results which we obtained are better than the known results. Finally, we give some applications to trapezoidal formula and to special means.

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Some new Hermite–Hadamard type inequalities for s-convex functions and their applications

Let ∥·∥ be a unitarily invariant norm on matrices. For matrices A,B,X with A,B positive semidefinite and X arbitrary, we prove that the function t↦∥|AtXB1−t|r∥·∥|A1−tXBt|r∥ is convex on [0,1] for each r>0. This convexity result interpolates the matrix Cauchy–Schwarz inequality ∥|A1/2XB1/2|r∥2⩽∥|AX|r∥·∥|XB|r∥ due to R. Bhatia and C. Davis [Linear Algebra Appl. 223/224 (1995) 119], and also it generalizes A.W. Marshall and I. Olkin's [Pacific J. Math. 15 (1965) 241] result that the condition number ∥As∥·∥A−s∥ is increasing in s>0. We prove that if f(t) is a nonnegative operator monotone function on [0,∞) and ∥·∥ is a normalized unitarily invariant norm, then f(∥X∥)⩽∥f(|X|)∥ for every matrix X. The special case when f(t)=tr (0<r⩽1) is used to consider the monotonicity of p↦∥Ap+Bp∥1/p as well as p↦∥(Ap+Bp)1/p∥. Furthermore, we obtain some norm inequalities of Holder and Minkowski types related to the expression ∥|A|p+|B|p∥1/p. For example, comparisons are made between ∥C*A+D*B∥ and ∥|A|p+|B|p∥1/p·∥|C|q+|D|q∥1/q, where p−1+q−1=1.

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Inequalities Involving Unitarily Invariant Norms and Operator Monotone Functions (Current topics on operator theory and operator inequalities)

Let be a factor von Neumann algebra and be the *-Jordan derivation on , that is, for every , where , then is additive *-derivation.

Non-linear *-Jordan derivations on von Neumann algebras

Abstract Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and 𝓐 ⊆ B(H) be a von Neumann algebra with no central summands of type I1. For A, B ∈ 𝓐, define by A ∙ B = AB+BA∗ a new product of A and B. In this article, it is proved that a map Φ: 𝓐 → B(H) satisfies Φ(A ∙ B ∙ C) = Φ(A) ∙ B ∙ C+A ∙ Φ(B) ∙ C+A ∙ B ∙Φ(C) for all A, B,C ∈ 𝓐 if and only if Φ is an additive *-derivation.

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

In this paper, we introduce the concept of operator geometrically convex functions for positive linear operators and prove some Hermite–Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results.

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Hermite–Hadamard type inequalities for operator geometrically convex functions

Additivity of maps preserving products AP ± PA* on C*-algebras

We introduce the concept of operator h-convex functions for positive linear maps, and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain several trace inequalities for operators.

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Some inequalities associated with the Hermite-Hadamard inequalities for operator h-convex functions

In this paper, we obtain some Berezin number inequalities based on the\n definition of Berezin symbol. Among other inequalities, we show that if A,B\n be positive definite operators in B(H), and A#B is the geometric mean of\n them, then ber2(A#B) ? ber (A2+B2/2)- 1/2 inf ????(?k); where ?(?k?)\n = ?(A-B)?k?,?k??2, and ?k? is the normalized reproducing kernel of the\n space H for ? belong to some set.

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Vahid Darvish

Papers

Non-linear *-Jordan derivations on von Neumann algebras

Hermite–Hadamard type inequalities for operator geometrically convex functions

Additivity of maps preserving products AP ± PA* on C*-algebras

Some inequalities associated with the Hermite-Hadamard inequalities for operator h-convex functions

Some upper bounds for the Berezin number of Hilbert space operators