• Corpus ID: 252519408

Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions

@inproceedings{Allu2022ProofOT,
  title={Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions},
  author={Vasudevarao Allu and Abhishek Pandey},
  year={2022}
}
. Let S denote the class of analytic and univalent ( i.e. , one-to-one) functions f ( z ) = z + P ∞ n =2 a n z n in the unit disk D = { z ∈ C : | z | < 1 } . For f ∈ S , Ma proposed the generalized Zalcman conjecture that | a n a m − a n + m − 1 | ≤ ( n − 1)( m − 1) , for n ≥ 2 , m ≥ 2 , with equality only for the Koebe function k ( z ) = z/ (1 − z ) 2 and its rotations. In this paper using the properties of holomorphic motion and Krushkal’s Surgery Lemma [12], we prove the generalized Zalcman… 
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References

SHOWING 1-10 OF 20 REFERENCES

Generalized Zalcman Conjecture for Starlike and Typically Real Functions

Abstract Zalcman conjectured that |a2n − a2n − 1| ≤ (n − 1)2,n = 2, 3,… forf(z) = z + a2z2 + a3z3 + ··· ∈ S, the class of normalized holomorphic and univalent functionsf(z) in the unit disk D . We…

The Zalcman conjecture for close-to-convex functions

Let S be the class of functions f (z) = z + * analytic and univalent in the unit disk D. For f (z) = z + a2z2 + * * * E S, Zalcman conjectured that la2 a2n-11 4 and close-to-convex functions. Let S…

Coefficients of meromorphic schlicht functions

This paper presents an elementary proof of a known theorem on the coefficients of meromorphic schlicht functions: if fG Ezand bk = Ofor 1 _k 1 except for a simple pole at oo with residue 1. Let 2o be…

Proof of the Zalcman conjecture for initial coefficients

Abstract The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions on the unit disk satisfy the inequality for all n > 2, with…

A uniqueness theorem for Beltrami equations

( 1 . 2 ) (z) -q(Z) (I Zz) 2, where q is analytic and subject to the condition (1.3) 1 q(t -Z)21 < k< 1. Elementary classical methods yield a particular solution of (t.1). It is the purpose of this…

On a coefficient problem for schlicht functions

1. The problem of maximizing la 3ka$1 within the class S was solved for k m-i 2m ' m = 1,2,..., by Fekete and Szeg~ in 1933, and later for all real k by J.A. Jenkins [2]. We consider here the problem…

Holomorphic families of injections

will be called admissible if riO, z)=z for all z EE, for every fixed 2 EAr the map f(2, .):E-->(~ is an injection, and for every fixed zEE the map f(. ,z): Ar--~C is holomorphic (i.e., a meromorphic…

Holomorphic motions and polynomial hulls

A holomorphic motion of E C C over the unit disc D is a map f: DxC-?C such that f(O, w) =w, w E E, the function f(z, w) = fz(w) is holomorphic in z, and fz: EC is an injection for all z E D.…

Univalent functions and Teichm?uller space

I Quasiconformal Mappings.- to Chapter I.- 1. Conformal Invariants.- 1.1 Hyperbolic metric.- 1.2 Module of a quadrilateral.- 1.3 Length-area method.- 1.4 Rengel's inequality.- 1.5 Module of a ring…

Univalent functions and holomorphic motions

We consider in this paper the coefficient problems for univalent functions slightly generalizing the Bieberbach and Zalcman conjectures and give their complete solution for the lower coefficients.…