A Polya "shire" theorem for functions with algebraic singularities.
A polynomial with 2k critical values at infinity
We construct a polynomial f:ℂ² → ℂ of degree 4k+2 with no critical points in ℂ² and with 2k critical values at infinity.
A power of a meromorphic function sharing two small functions with a derivative of the power
In connection to a conjecture of W. Lü, Q. Li and C. Yang (2014), we prove a result on small function sharing by a power of a meromorphic function with few poles with a derivative of the power. Our results improve a number of known results.
A preserving property of a Bernardi type operator.
A preserving property of generalized Libera integral operator.
A preserving property of the Alexander operator.
A preserving property of the generalized Bernardi integral operator.
A probabilistic description of the three theorems of Hardy.
A Problem Concerning the Zeros of a Certain Kind of Holomorphic Function in the Unit Disk.
A problem of linear conjugation for analytic functions with boundary values from the Zygmund class.
A proof of Euler's identity by a functional equation.
A proof of Ilieff's conjecture for polynomials with four zeros.
A Proof of Simultaneous Linearization with a Polylog Estimate
We give an alternative proof of simultaneous linearization recently shown by T. Ueda, which connects the Schröder equation and the Abel equation analytically. In fact, we generalize Ueda's original result so that we may apply it to the parabolic fixed points with multiple petals. As an application, we show a continuity result on linearizing coordinates in complex dynamics.
A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions
Let D denote the open unit disc and f:D → ℂ̅ be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0,1) and an expansion , |z| < p. In particular, we consider f that map D onto a domain whose complement with respect to ℂ̅ is convex. Because of the shape of f(D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for p ∈ (0,1) the domain of variability of the coefficient...
A proper subclass of MacLane's class .
A property of entire transcendental functions
A property of the class of functions whose derivative has a positive real part.
A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres
A hyperbolic geodesic joining two punctures on a Riemann surface has infinite length. To obtain a useful distance-like quantity we define a finite pseudo-length of such a geodesic in terms of the hyperbolic length of its surrounding geodesic loop. There is a well defined angle between two geodesics meeting at a puncture, and our pseudo-trigonometry connects these angles with pseudo-lengths. We state and prove a theorem resembling Ptolemy's classical theorem on cyclic quadrilaterals and three general...
A pure smoothness condition for Radó’s theorem for -analytic functions
Let be a bounded, simply connected -convex domain. Let and let be a function on which is separately -smooth with respect to (by which we mean jointly -smooth with respect to , ). If is -analytic on , then is -analytic on . The result is well-known for the case , , even when a priori is only known to be continuous.
A purely number theoretic attempt to prove Picard's theorems