Johnny E. Brown

Quasiconformal extensions for some geometric subclasses of univalent functions

Let S denote the set of all functions f which are analytic and univalent in the unit disk D normalized so that f(z)=z

On a coefficient problem for nonvanishing H p functions

It has been conjectured by Hummel, Scheinberg and Zalcman that if f(z) ≈ a 0 + a 1 z + a 2 z 2 + … is a nonvanishing H p function with then for all n⋝ 1 and 1 < p < ∞ where 1/p + 1/q = 1. Using the Pontryagin Maximum Principle we solve a related extremal problem which proves the conjecture in the case n = 1 and in the case of arbitrary n ⋝ 2 provided a m = 0 for 1 m < (n 1)/2.

A sequence of extremal problems for trigonometric polynomials

The sequence of extremal problems In = sup{(2π)−1 ∝02π¦p(θ)¦2 dθ¦pϵ Pn}, where Pn denotes the set of nonnegative trigonometric polynomials of degree ⩽n having constant term 1, is studied. It is shown that (n + 1) C1 ⩽ In < 1 + (n + 1) C1, where C1 = 0.686981293….

Iteration of functions subordinate to schlicht functions

For a fixed schlicht function G using iterations we find necessary and sufficient conditions on the coefficients of a function f so that f is subordinate to G. We also show how these iterations may be used to obtain sharp coefficient inequalities for such functions and apply the results to the Krzyz conjecture for 1 ≤ n ≤ 4.

SOME SHARP NEIGHBORHOODS OF UNIVALENT FUNCTIONS

For 8 > 0 and f(z) = z + a2z2 + * * * analytic in IzI 0 we define the 8-neighborhood of f by

Lp-norms of polynomials with positive real part

We derive an estimate for Δn, 1 = sup{(2π)−1 ∝02π¦p(eit)¦dt: p(z) = 1 + a1z + · · · + anzn, Re(p(z)) > 0 for ¦z¦ < 1}. In particular it is shown that Δn, 1 ⩽ 1 + log(C1(n + 1) + 1), where C1 = 0.686981293…, It is also shown that 2π ⩽ lim infn → ∞ Δn, 1log n. Finally, upper bounds are found for the Lp-norms of polynomials with positive real part on the unit disk.

A method for investigating geometric properties of support points and applications

A normalized univalent function f is a support point of S if there exists a continuous linear functional L (which is nonconstant on S) for which f maximizes Re L(g), g E S. For such functions it is known that r = C f(U) is a single analytic arc that is part of a trajectory of a certain quadratic differential Q(w) dw2. A method is developed which is used to study geometric properties of support points. This method depends on consideration of Jm{w2Q(w)} rather than the usual Re{w2Q(w)}. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.