Generalizations of a parameterized Jordan-type inequality,Janous’s inequality and Tsintsifas’s inequality

In this work, we first prove a generalized version of a parameterized Jordan-type inequality. We then use it to prove the generalized versions of Janous’s inequality and Tsintsifas’s inequality which reduce to two inequalities conjectured by Janous and Tsintsifas as special cases.     

On Jordan’s inequality

We present sharp upper and lower bounds for the function \(\sin (x)/x\). Our bounds are polynomials of degree 2n, where n is any nonnegative integer.     

On Bohr’s inequality

It is established that H. Bohr’s inequality \(\sum\nolimits_{k = 0}^\infty {\left| {{{f^{\left( k \right)} \left( 0 \right)} \mathord{\left/ {\vphantom {{f^{\left( k \right)} \left( 0 \right)} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right| \leqslant \sqrt 2 \left\| f \right\|_\infty }\) is sharp on the class H _∞.     

Hlawka’s functional inequality

The paper is devoted to the functional inequality (called by us Hlawka’s functional inequality) $$f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z)$$ for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality: $$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$ which in particular holds true for all x, y, z from a real or complex inner product space.     

On Jensen-McShane’s inequality

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.     

On Hasse’s inequality

We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.     

Generalizations of Sherman’s inequality

The concept of majorization is a powerful and useful tool which arises frequently in many different areas of research. Together with the concept of Schur-convexity it gives an important characterization of convex functions. The well known Majorization theorem plays a very important role in majorization theory—it gives a relation between one-dimensional convex functions and n-dimensional Schur-convex functions. A more general result was obtained by S. Sherman. In this paper, we get generalizations of these results for n-convex functions using Taylor’s interpolating polynomial and the ?eby?ev functional. We apply the exponentially convex method in order to interpret our results in the form of exponentially, and in the special case logarithmically convex functions. The outcome is some new classes of two-parameter Cauchy-type means.     

Sharpening Jordan’s inequality and the Yang Le inequality

In this work, the following inequality: $\frac{sinx}{x}\le \frac{2}{\pi }+\frac{\pi -2}{{\pi }^3}({\pi }^2-4x^2)\text{,}x\in (0,\pi /2]$ is established. An application of this inequality gives an improvement of the Yang Le inequality [C.J. Zhao, Generalization and strengthening of the Yang Le inequality, Math. Practice Theory 30 (4) (2000) 493–497 (in Chinese)]: where $A_i>0(i=1,2,\dots ,n),{\sum }_{i=1}^n{A}_i\le \pi ,0\le \lambda \le 1$, and $n\ge 2$ is a natural number.     

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.     

Jensen’s inequality on convex spaces

We prove a Jensen’s inequality on $p$ -uniformly convex space in terms of $p$ -barycenters of probability measures with $(p-1)$ -th moment with $p\in ]1,\infty [$ under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT $(\kappa )$ -space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into $2$ -uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.     

Korn’s inequality and John domains

Let \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\), be a bounded domain satisfying the separation property. We show that the following conditions are equivalent:

1. (i)
   \(\Omega \) is a John domain;
    
2. (ii)
   for a fixed \(p\in (1,\infty )\), the Korn inequality holds for each \(\mathbf {u}\in W^{1,p}(\Omega ,\mathbb {R}^n)\) satisfying \(\int _\Omega \frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\,dx=0\), \(1\le i,j\le n\),

   $$\begin{aligned} \Vert D\mathbf {u}\Vert _{L^p(\Omega )}\le C_K(\Omega , p)\Vert \epsilon (\mathbf {u})\Vert _{L^p(\Omega )}; \qquad (K_{p}) \end{aligned}$$
    
3. (ii’)
   for all \(p\in (1,\infty )\), \((K_p)\) holds on \(\Omega \);
    
4. (iii)
   for a fixed \(p\in (1,\infty )\), for each \(f\in L^p(\Omega )\) with vanishing mean value on \(\Omega \), there exists a solution \(\mathbf {v}\in W^{1,p}_0(\Omega ,\mathbb {R}^n)\) to the equation \(\mathrm {div}\,\mathbf {v}=f\) with

   $$\begin{aligned} \Vert \mathbf {v}\Vert _{W^{1,p}(\Omega ,\mathbb {R}^n)}\le C(\Omega , p)\Vert f\Vert _{L^p(\Omega )};\qquad (DE_p) \end{aligned}$$
    
5. (iii’)
   for all \(p\in (1,\infty )\), \((DE_p)\) holds on \(\Omega \).
    

For domains satisfying the separation property, in particular, for finitely connected domains in the plane, our result provides a geometric characterization of the Korn inequality, and gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal 217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013). For the plane, our result is best possible in the sense that, there exist infinitely connected domains which are not John but support Korn’s inequality.

     

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.     

Sharpened versions of a Kolmogorov’s inequality

Sharpened versions of a Kolmogorov’s inequality for sums of independent Bernoulli random variables are proved.     

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.     

A spinorial analogue of Aubin’s inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that

Minkowski’s inequality and sums of squares

Positive polynomials arising from Muirhead’s inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski’s inequality can be rewritten as sums of squares.     

The error term in Nevanlinna’s inequality

An upper bound is given for the error termS(r, |a _j |,f) in Nevanlinna’s inequality. For given positive increasing functions p and $ with ∫ ₁ ^∞ dr/p(r) = ∫ ₁ ^∞ dr/r ?(r) = ∞, setP(r) = ∫ ₁ ^r dt/p,Ψ(r) = ∫ ₁ ^r dt/t ?(t) We prove that $$S(r, \left\{ {a_j } \right\}, f) \leqslant \log \frac{{T(r, f)\phi (T(r, f))}}{{p(r)}} + O(1)$$ holds, with a small exceptional set of r, for any finite set of points |a _j | in the extended plane and any meromorphic function f such thatΨ(T(r, f)) =O(P(r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered.