Around Jensen’s inequality for strongly convex functions

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \), then

$$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$

and if \(Sp\left( A \right) \subset \left( 0,1 \right) \), then

$$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$

for each positive operator A and \(x\in \mathcal {H}\) with \(\left\| x \right\| =1\).

     

A note on Jensen’s inequality for BSDEs

Under the Lipschitz assumption and square integrable assumption on g, Jiang proved that Jensen＇s inequality for BSDEs with generator g holds in general if and only if g is independent of y, g is super homogenous in z and g（t, 0） = 0, a.s., a.e.. In this paper, based on Jiang＇s results, under the same assumptions as Jiang＇s, we investigate the necessary and sufficient condition on g under which Jensen＇s inequality for BSDEs with generator g holds for some specific convex functions, which generalizes some known results on Jensen＇s inequality for BSDEs.     

On Jensen’s inequality and Hölder’s inequality for g-expectation

In this paper, we shall prove that for n > 1, the n-dimensional Jensen inequality holds for the g-expectation if and only if g is independent of y and linear with respect to z, in other words, the corresponding g-expectation must be linear. A Similar result also holds for the general nonlinear expectation defined in Coquet et al. (Prob. Theory Relat. Fields 123 (2002), 1–27 or Peng (Stochastic Methods in Finance Lectures, LNM 1856, 143–217, Springer-Verlag, Berlin, 2004). As an application of a special n-dimensional Jensen inequality for g-expectation, we give a sufficient condition for g under which the Hölder’s inequality and Minkowski’s inequality for the corresponding g-expectation hold true.     

Jensen’s inequality and new entropy bounds

We establish new lower and upper bounds for Jensen’s discrete inequality. Applying those results in information theory, we obtain new and more precise bounds for Shannon’s entropy.     

Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces

We introduce the weak Hardy-Morrey spaces in this paper. We also obtain the atomic decompositions of the weak Hardy-Morrey spaces. By using these decompositions, we establish the Hardy inequalities on the weak Hardy-Morrey spaces.     

Jensen’s inequality for g-convex function under g-expectation

A real valued function h defined on ${\mathbb{R}}$ is called g-convex if it satisfies the “generalized Jensen’s inequality” for a given g-expectation, i.e., ${h(\mathbb{E}^{g}[X])\leq \mathbb{E}^{g}[h(X)]}$ holds for all random variables X such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient condition for a C ²-function being g-convex, and study some more general situations. We also study g-concave and g-affine functions, and a relation between g-convexity and backward stochastic viability property.     

Wolff’s inequality in multi-parameter Morrey spaces

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L ^p (R ^d ) and multi-parameter Morrey spaces ${L^p_\lambda (R^d)}$ , where ${R^d=R^{n_1} \times R^{n_2} \times \cdots \times R^{n_k},\, \lambda = (\lambda _1,\ldots ,\lambda _k})$ and 0?<?λ _i ≤ n _i , 1?≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.     

The application of multi-dimensional Jensen’s inequality for G-martingale

Jensen’s inequalities play a key role in probability theory. In this paper, we aim to construct G-martingales by Jensen’s inequalities in the form of G-expectation, and extend the above results to the n-dimensional case.     

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.     

Refinements of Fejér’s inequality for convex functions

In this paper, we establish some new refinements for the celebrated Fejér??s and Hermite-Hadamard??s integral inequalities for convex functions.     

Short note on Hilbert’s inequality

Considering five different parameters, we obtain some new Hilbert-type integral inequalities for functions f(x), g(x) in L²[0, ∞). Then, we extract from our results some special cases which have been proved before.     

Necessary and sufficient conditions for the validity of Jensen’s inequality

We consider the d-dimensional Jensen inequality $$ T[\varphi(f_1, \dots, f_d)]\, \ge \, \varphi(T[f_1], \dots, T[f_d])\quad\quad(\ast)$$ T [ φ ( f 1 , … , f d ) ] ≥ φ ( T [ f 1 ] , … , T [ f d ] ) ( * ) as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set ${K\subset \mathbb{R}^d}$ K ? R d , and f ₁, . . . , f _d are from some linear space of functions. Our aim is to find necessary and sufficient conditions for the validity of (*). In particular, we show that if we exclude three types of convex sets K, then Jensen’s inequality holds for a sublinear functional T if and only if T is linear, positive, and satisfies T[1] = 1. Furthermore, for each of the excluded types of convex sets, we present nonlinear, sublinear functionals T for which Jensen’s inequality holds. Thus the conditions on K are optimal. Our contributions generalize or complete several known results.     

Korpelevich’s method for variational inequality problems in Banach spaces

We propose a variant of Korpelevich’s method for solving variational inequality problems with operators in Banach spaces. A full convergence analysis of the method is presented under reasonable assumptions on the problem data.     

Newton’s method on generalized Banach spaces

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.     

A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality

We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand’s transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry–Emery curvature is not lower bounded. Several new examples are provided.     

Notes on Schneider’s stability estimates for convex sets

In 2009 Schneider obtained stability estimates in terms of the Banach–Mazur distance for several geometric inequalities for convex bodies in an n-dimensional normed space ${\mathbb{E}^n}$ . A unique feature of his approach is to express fundamental geometric quantities in terms of a single function ${\rho:\mathfrak{B} \times \mathfrak{B} \to \mathbb{R}}$ defined on the family of all convex bodies ${\mathfrak{B}}$ in ${\mathbb{E}^n}$ . In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric d _D on ${\mathfrak{B}}$ inducing, from our point of view, a finer topology than Banach–Mazur’s d _BM . Further, d _D induces a metric on the quotient ${\mathfrak{B}/{\rm Dil}^+}$ of ${\mathfrak{B}}$ by the relation of positive dilatation (homothety). Unlike its compact Banach–Mazur counterpart, d _D is only “boundedly compact,” in particular, complete and locally compact. The general linear group ${{\rm GL}(\mathbb{E}^n)}$ acts on ${\mathfrak{B}/{\rm Dil}^+}$ by isometries with respect to d _D , and the orbit space is naturally identified with the Banach–Mazur compactum ${\mathfrak{B}/{\rm Aff}}$ via the natural projection ${\pi:\mathfrak{B}/{\rm Dil}^+\to\mathfrak{B}/{\rm Aff}}$ , where Aff is the affine group of ${\mathbb{E}^n}$ . The metric d _D has the advantage that many geometric quantities are explicitly computable. We show that d _D provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of Schneider turn out to be valid with d _BM replaced by d _D .     

Analog of Borsuk’s problem on Banach spaces

An analog of the well-known problem of partition of figures into parts with less diameter in Banach spaces is studied. The sufficient conditions for the sets to belong to the class of Borsuk’s sets in multidimensional Banach spaces are first obtained.