Inequalities For Sequences Research Articles

Higher order Turán inequalities for combinatorial sequences

The Turán inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. It is well known that if a real entire function ψ(x) is in the LP class, the Maclaurin coefficients satisfy both the Turán inequality and the higher order Turán inequality. Chen, Jia and Wang proved that for n≥95, the higher order Turán inequality holds for the partition function p(n) and the 3-rd associated Jensen polynomials p(n)+3p(n+1)x+3p(n+2)x2+p(n+3)x3 have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to ξ(s) and the partition function, are hyperbolic for sufficiently large n. This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turán inequality for the sequences {an/n!}n≥0, where an satisfy a three-term recurrence relation. In particular, we prove higher order Turán inequality for the sequences {an/n!}n≥0, where an are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomialsann!+3an+1(n+1)!x+3an+2(n+2)!x2+an+3(n+3)!x3 have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer m≥4, there exists an integer N(m) such that for n>N(m), the m-th associated Jensen polynomials∑i=0m(mi)an+i(n+i)!xi have only real zeros.

Hermitian Forms and Inequalities for Sequences and Power Series of Operators in Hilbert Spaces

AbstractBy the use of some inequalities for nonnegative Hermitian forms various inequalities for sequences and power series of bounded linear operators in complex Hilbert spaces are established. Applications for some fundamental functions of interest are also given.

Further Inequalities for Sequences and Power Series of Operators in Hilbert Spaces Via Hermitian Forms

AbstractBy the use of some inequalities for nonnegative Hermitian forms some new inequalities for sequences and power series of bounded linear operators in complex Hilbert spaces are established. Applications for some fundamental functions of interest are also given.

Mixed norm estimates for the Riesz transforms on the Heisenberg group

In this paper we prove weighted mixed norm estimates for Riesz transforms on the Heisenberg group and Riesz transforms associated to the special Hermite operator. From these results vector-valued inequalities for sequences of Riesz transforms associated to generalised Grushin operators and Laguerre operators are deduced.

Another Grüss Type Inequality for Sequences of Vectors in Normed Linear Spaces and Applications

A discrete inequality of Gruss type in normed linear spaces and applications for the discrete Fourier transform, Mellin transform of sequences, for polynomials with coefficients in normed spaces and for vector valued Lipschitzian mappings are given.

On Chebyshev's inequality for sequences

We give improved versions of Chebyshev's inequality for star-shaped sequences, valid also for convex sequences.

Bounds for the operator norms of some Nörlund matrices

Suppose ( p n ) n ≥ 0 (p_n)_{n \geq 0} is a non-increasing sequence of non-negative numbers with p 0 = 1 p_0 = 1 , P n = ∑ j = 0 n p j P_n = \sum _{j=0}^n p_j , n = 0 , 1 … n = 0, 1 \dots , and A = A ( p n ) = ( a n k ) A = A(p_n) = (a_{nk}) is the lower triangular matrix defined by a n k = p n − k / P n a_{nk} = p_{n-k} / P_n , 0 ≤ k ≤ n 0 \leq k \leq n , and a n k = 0 a_{nk} = 0 , n > k n > k . We show that the operator norm of A A as a linear operator on ℓ p \ell _p is no greater than p / ( p − 1 ) p / (p-1) , for 1 > p > ∞ 1 > p > \infty ; this generalizes, yet again, Hardy’s inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the p n p_n tend to a positive limit, the operator norm of A A on ℓ p \ell _p is exactly p / ( p − 1 ) p/(p-1) . We also give some cases when the operator norm of A A on ℓ p \ell _p is less than p / ( p − 1 ) p/(p-1) .

Korn’s inequality for sequences

A version of the Korn’s inequality valid for a weakly convergent sequence of deformations is given.