ON ZYGMUND–TYPE INEQUALITIES CONCERNING POLAR DERIVATIVE OF POLYNOMIALS

Let \(P(z)\) be a polynomial of degree \(n\), then concerning the estimate for maximum of \(|P'(z)|\) on the unit circle, it was proved by S. Bernstein that \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension of this inequality. The polar derivative \(D_{\alpha}[P](z)\) of \(P(z)\), with respect to a point \(\alpha \in \mathbb{C}\), generalizes the ordinary derivative in the sense that \(\lim_{\alpha\to\infty} D_{\alpha}[P](z)/{\alpha} = P'(z).\) Recently, for polynomials of the form \(P(z) = a_0 + \sum_{j=\mu}^n a_jz^j,\) \(1\leq\mu\leq n\) and having no zero in \(|z| < k\) where \(k > 1\), the following Zygmund-type inequality for polar derivative of \(P(z)\) was obtained: $$\|D_{\alpha}[P]\|_p\leq n \Big(\dfrac{|\alpha|+k^{\mu}}{\|k^{\mu}+z\|_p}\Big)\|P\|_p, \quad \text{where}\quad |\alpha|\geq1,\quad p>0.$$In this paper, we obtained a refinement of this inequality by involving minimum modulus of \(|P(z)|\) on \(|z| = k\), which also includes improvements of some inequalities, for the derivative of a polynomial with restricted zeros as well.

Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

For a transcendental entire function f, the property that there exists $$r>0$$ r > 0 such that $$m^n(r)\rightarrow \infty $$ m n ( r ) → ∞ as $$n\rightarrow \infty $$ n → ∞ , where $$m(r)=\min \{|f(z)|:|z|=r\}$$ m ( r ) = min { | f ( z ) | : | z | = r } , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

On Subharmonic and Entire Functions of Small Order: After Kjellberg

We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.

Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

We consider the class of real transcendental entire functions $f$ of finite order with only real zeros and show that if the iterated minimum modulus tends to $\infty $, then the escaping set $I(\,f)$ of $f$ has the structure of a spider’s web, in which case Eremenko’s conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko’s conjecture. For functions in this class, we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function $f$.

Division algebras of slice functions

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

APPROXIMATELY ANGLE PRESERVING MAPPINGS

We study linear mappings which preserve vectors at a specific angle. We introduce the concept of $(\unicode[STIX]{x1D700},c)$ -angle preserving mappings and define $\widehat{\unicode[STIX]{x1D700}}\,(T,c)$ as the ‘smallest’ number $\unicode[STIX]{x1D700}$ for which $T$ is an $(\unicode[STIX]{x1D700},c)$ -angle preserving mapping. We derive an exact formula for $\widehat{\unicode[STIX]{x1D700}}\,(T,c)$ in terms of the norm $\Vert T\Vert$ and the minimum modulus $[T]$ of $T$ . Finally, we characterise approximately angle preserving mappings.

Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

In this paper, we obtain new sufficient conditions of boundedness of L-index in joint variables for entire function in Cn functions. They give an estimate of maximum modulus of an entire function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives and the distribution of zeros. In some sense, the obtained results are new for entire functions of bounded index and l-index in C too. They generalize known results of Fricke, Sheremeta, and Kuzyk.