On the univalence of derivatives of functions which are univalent in angular domains

We consider functions f that are univalent in a plane angular domain of angle απ, 0 < α ≤ 2. It is proved that there exists a natural number k depending only on α such that the kth derivatives f ^(k) of these functions cannot be univalent in this angle. We find the least of the possible values of for k. As a consequence, we obtain an answer to the question posed by Kir’yatskii: if f is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.     

On the univalence of the log-biharmonic mappings

In this paper, we establish the univalence and starlikeness connection between log-biharmonic mappings and logharmonic mappings.     

On the univalence of an integral operator

In this work the author introduces a general integral operator and determines conditions for the univalence of this integral operator.     

On univalence and P-matrices

Suppose F is a differentiable mapping from a rectangle R?Eⁿ into Eⁿ. Gale and Nikaido proved that if the Jacobian of F is a P-matrix in R, then F is univalent in R. Their paper has served as the basis of numerous results on univalence. Recently H. Scarf conjectured a significant extension: that the Jacobian of F need not be a P-matrix everywhere in the rectangle R, but merely on its boundary. This paper proves Scarf's conjecture, and to do so utilizes a conceptually different method of proof than that of Gale and Nikaido. The proof is presented in such a way as to demonstrate a suggestion of Scarf that orientation arguments may provide an alternative proof of the Gale-Nikaido theorem.     

On the Radii of univalence of Gel'fond-Leont'ev derivatives

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 390–399, March, 1995.     

On univalence of two integral operators

In this work we consider two integral operators. The integral operators were constructed on the basis of the fact that the number of functions from the composition of the operators is the entire part of a complex number modulus.     

On the univalence of functions with logharmonic laplacian

In this paper, we prove Radó’s theorem holds for functions of the form is logharmonic. We show that if F is of the form , where is logharmonic, then F is starlike iff ψ(z)=h(z)/g(z) is starlike. In addition, when , where L is logharmonic and H is harmonic, we give the sufficient conditions for F to be locally univalent.     

On some criteria for the univalence of analytic functions

Criteria for the univalence of functions f(z) which are regular in the domain ¦z¦ < 1 are obtained in the form of restrictions on the modulus of the quotient f (z) [f '(z)]^–1. Analogous results are obtained also for functions which are analytic in the interior of the unit disk except for a simple pole at infinity.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 359–366, March, 1973.In conclusion the author wishes to thank L. A. Aksent'ev and F. G. Avkhadiev for discussion of the results and valuable remarks.     

On the radius of univalence of a polynomial

Mathematische Zeitschrift -     

On the univalence of functions defined by certain integral transforms

The integral transform $\text{F(z) = ∝}{}_0{}^{\mathrm{z}}\text{(f′(t))}{}^{\mathrm{\alpha }}\text{(}\text{g(t)}\text{t}\text{)}{}^{\mathrm{\beta }}\text{dt}$, where α and β are real, of pairs of special analytic functions f(z) = z + ···, g(z) = z + ···, univalent in the open unit disc Δ is studied. The transform and our results extend some recent results due to Shirakova.     

On the univalence of an integral operator defined by Hadamard product

In this paper, we introduce a new general integral operator defined by the Hadamard product. Furthermore, we obtained new sufficient conditions for this operator to be univalent in the open unit disk.     

On the Gale-NikaidÔ-Inada fundamental theorem on univalence of mappings

The research was supported by the Russian Foundation for Fundamental Research (Grant 93-011-228)