By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

Abstract. The weighted mean matrix ${{M}_{a}}$ is the triangular matrix $\left\{ {{a}_{k}}/{{A}_{n}} \right\}$, where ${{a}_{n}}\,>\,0$ and ${{A}_{n}}\,:=\,{{a}_{1}}\,+\,{{a}_{2}}\,+\cdots +\,{{a}_{n}}$. It is proved that, subject to ${{n}^{c}}{{a}_{n}}$ being eventually monotonic for each constant $c$ and to the existence of $\alpha \,:=\,\lim \,\frac{{{A}_{n}}}{n{{a}_{n}}},\,{{M}_{a}}\,\in \,B\left( {{l}_{p}} \right)$ for $1\,<\,p\,<\infty $ if and only if $\alpha \,<\,p$.