Narrow Results

For a sequence ${\{M_n\}}_{n=1}^{\infty }$ of expanding matrices with $M_n\in M_d(\mathbb{Z})$ and a sequence ${\{D_n\}}_{n=1}^{\infty }$ of finite digit sets with $D_n\subset {\mathbb{Z}}^d$, the Moran measure ${\mu }_{\{M_n\},\{D_n\}}$ is defined by the infinite convolution

Yıldırım obtained an asymptotic formula of the discrete moment of $|\zeta (\frac{1}{2}+it)|$ over the zero of the higher derivatives of Hardy’s $Z$-function. We give a generalization of his result on Hardy’s $Z$-function.

We count the number of critical points of a modular form with real Fourier coefficients in a $\gamma$-translate of the standard fundamental domain ${ℱ}$ (with $\gamma \in {\mathrm{\text{SL}}}_2(\mathbb{Z})$). Whereas by the valence formula the (weighted) number of zeros of this modular form in $\gamma {ℱ}$ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of ${ℱ}$, the value of ${\gamma }^{-1}(\infty )$ and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form.

This paper considers the well-known Erdös–Lax and Turán-type inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. The obtained results produce inequalities that are sharper than the previous ones. Moreover, a numerical example is presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.

In this paper, some generalizations of lower bound estimates for the maximum modulus of the $t$th polar derivative $D_{{\xi }_t}{D}_{{\xi }_{t-1}}\dots D_{{\xi }_2}{D}_{{\xi }_1}$, where ${\xi }_t\in ℂ$, $1\le t<n$ of polynomials are established under the assumption that one of the underlying polynomials $p$ and $q$ does not vanish in $\left|z\right|>k,k\le 1$. The obtained estimates include several known lower bounds for polar derivatives as special cases.