For triangular arrays $\{X_{n,k}:1\leqslant k\leqslant n,n\geqslant 1\}$ of upper extended negatively dependent random variables weakly mean dominated by a random variable $X$ and sequences $\{b_{n}\}$ of positive constants, conditions are given to guarantee an almost sure finite upper bound to $\sum _{k=1}^{n}(X_{n,k}-\mathbb{E}X_{n,k})/\!\sqrt{b_{n}\,\text{Log}\,n}$, where $\text{Log}\,n:=\max \{1,\log n\}$, thus getting control over the limiting rate in terms of the prescribed sequence $\{b_{n}\}$ and permitting us to weaken or strengthen the assumptions on the random variables.

The inequality of strong summability for the Marcinkiewicz means of multiply Fourier series is proved. The inequalities of strong summability with gaps for the different classes of integrable functions are established. The Bernstein inequality for the fractional derivative of analytic polynomials is proved.

If is an univariate polynomial with degree n then Bombieri norm of P is defined by

where denotes the binomial coefficient.

In the present paper we give, under assumptions on the roots of P, optimal Bernsteintype inequalities for the ratio between Bombieri norm of P and that of its derivative P′.

We also give such inequalities for the polar derivatives of P defined by

Generalized polynomials are defined as products of polynomials raised to positive real powers. The generalized degree can be introduced in a natural way. Several inequalities holding for ordinary polynomials are expected to be true for generalized polynomials, by utilizing the generalized degree in place of the ordinary one. Based on Remez-type inequalities on the size of generalized polynomials, we establish Bernstein and Markov type inequalities for generalized non-negative polynomials, obtaining the best possible result up to a multiplicative absolute constant.