Tusnady’s inequality is the key ingredient in the KMT/Hungarian coupling of the empirical distribution function with a Brownian Bridge. We present an elementary proof of a result that sharpens the Tusnady inequality, modulo constants. Our method uses the beta integral representation of binomial tails, simple Taylor expansion, and some novel bounds for the ratios of normal tail probabilities. Ke...

It is well known that differential equations, integral equations and integral-differential equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physical, mechanics, chemistry, aerodynamics, and the electrodynamics of complex mediums, etc. Gronwal...

In recent years many integral inequalities for continuous and discontinuous functions have been established, which provide handy tools for investigating the quantitative and qualitative properties of solutions of integral and differential equations [1-17]. In the investigations for integral inequalities, the idea of generalizing known integral inequalities have gained extensive attention. In [1...

It is shown that maximal truncations of nonconvolution L-bounded singular integral operators with kernels satisfying Hörmander’s condition are weak type (1, 1) and L bounded for 1 < p < ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the bounded...

X iv :m at h/ 04 06 10 8v 1 [ m at h. FA ] 6 J un 2 00 4 QUADRATIC REVERSES OF THE CONTINUOUS TRIANGLE INEQUALITY FOR BOCHNER INTEGRAL OF VECTOR-VALUED FUNCTIONS IN HILBERT SPACES SEVER S. DRAGOMIR Abstract. Some quadratic reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are prov...

In this paper, we define a class R h p,m,β associated with convolution of p-valent analytic functions. Some properties in the form of coefficient inequality, growth and distortion bounds, sufficient conditions with the help of various lemmas, integral means inequality for convolution of two functions and a set of class preserving integral operators of functions belonging to this class are studi...

In the article, a functional inequality in abstract spaces is established, which gives a new affirmative answer to an open problem posed by Feng Qi in [9]. Moreover, some integral inequalities and a discrete inequality involving sums are deduced.

Let x : M → Sn+p be an n-dimensional submanifold in an (n + p)dimensional unit sphere Sn+p, x : M → Sn+p is called a Willmore submanifold if it is an extremal submanifold to the following Willmore functional: ∫ M (S − nH) 2 dv, where S = ∑ α,i,j (hij) 2 is the square of the length of the second fundamental form, H is the mean curvature of M . In [13], author proved an integral inequality of Sim...

In this paper, we consider the mixed nonlinear hyperbolic equations with forcing term via Riccati inequality. Some sufficient conditions for the oscillation are derived by using Young's inequality and integral averaging method.

Some new refinements of the Schwarz inequality in inner product spaces are given. Applications for discrete and integral inequalities including the Heisenberg inequality for vectorvalued functions in Hilbert spaces are provided.