We create two Sakaguchi-type function classes that are starlike and convex with respect to their symmetric points, including a q-difference operator, which may have or assymetric properties, in the open unit disc. first obtain sufficient coefficient bounds for these functions. In view of bounds, we quasi-Hadamard products several partial sums classes. Moreover, special values parameters provide...

Let A and B be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of p does the majorisation σ(BpAp) ≺w σ((BA) p) hold, and for which values its reversed inequality σ(BpAp) ≻w σ((BA) p).

We give a short proof of the Cwikel–Lieb–Rozenblum (CLR) bound on the number of negative eigenvalues of Schrödinger operators. The argument, which is based on work of Rumin, leads to remarkably good constants and applies to the case of operator-valued potentials as well. Moreover, we obtain the general form of Cwikel’s estimate about the singular values of operators of the form f(X)g(−i∇).

We use `p estimates together with Brascamp-Lieb inequalities to obtain bounds on the variance of the solution u" to the elliptic equation r" a(x="; ')r"u" = r" f . The variance is shown to be O("r) where the exponent r depends on the contrast of a(x; '). In two dimensions, we show that variance of the Greens function of the above elliptic operator is bounded. We also obtain decay rates for the ...

We consider the Brascamp–Lieb inequalities concerning multilinear integrals of products of functions in several dimensions. We give a complete treatment of the issues of finiteness of the constant, and of the existence and uniqueness of centred gaussian extremals. For arbitrary extremals we completely address the issue of existence, and partly address the issue of uniqueness. We also analyse th...

We prove global versions of certain known nonlinear Brascamp– Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp–Lieb estimates under such a homogeneity assumption.

Making use of convolution product, we introduce a unified class of analytic functions with negative coefficients. Also, we obtain the coefficient bounds, extreme points and radius of starlikeness for functions belonging to the generalized class TP ( , ). Furthermore, partial sums fk(z) of functions f(z) in the class P  ( , ) are considered and sharp lower bounds for the ratios of real part of ...

We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for Γ0(2). Numerical evidence indicates that a sharper bound holds for the weights h ≡ 2 ( mod 4). We derive upper bounds for the minimum positive integer represented by level two even positive-definite quadratic forms. Our data suggest that, ...