Estimates for the maximal modulus of rational functions with prescribed poles

In this paper, we obtain certain sharp estimates for the maximal modulus of a rational function with prescribed poles. The proofs of the obtained results are based on the new version of the Schwarz lemma for regular functions which was suggested by Osserman. The obtained results produce several inequalities for polynomials as well.

Generalized Differentiability of Continuous Functions

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

Microglobal Analysis

We study some Phragmén-Lindelöf type results, which were considered as extensions of maximal modulus theorem. We shall consider them from the viewpoint of Fourier analysis. Our analysis shows that the Phragmén-Lindelöf theorems can be regarded as the convexity of microglobal wave front sets. We prove such theorems for elliptic systems of equations with constant coefficients.