Texas Tech University |
Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 79409-1042 Voice: (806)742-2566 x 226 FAX: (806)742-1112 Email: kent.pearce@ttu.edu |

Abstracts

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**When is Tail Mean Estimation More Ecient than Tail Median? Answers and Implications for Quantitative Risk Management**" with R.W. Barnard and A.A. Trindade.We investigate the relative effciency of the empirical "tail median" vs. "tail mean" as estimators of location when the data can be modeled by an exponential power distribution (EPD), a flexible family of light-tailed densities. By considering appropriate probabilities so that the quantile of the untruncated EPD (tail median) and mean of the left-truncated EPD (tail mean) coincide, limiting results are established concerning the ratio of asymptotic variances of the corresponding estimators. The most remarkable finding is that in the limit of the right tail, the asymptotic variance of the tail median estimate is approximately 36% larger than that of the tail mean, irrespective of the EPD shape parameter. This discovery has important repercussions for quantitative risk management practice, where the tail median and tail mean correspond to value-at-risk and expected shortfall, respectively. To this effect, a methodology for choosing between the two risk measures that maximizes the precision of the estimate is proposed. From an extreme value theory perspective, analogous results and procedures are discussed also for the case when the data appear to be heavy-tailed.

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**Iceberg-Type Problems in Two Dimensions**" with R.W. Barnard and A.Y. Solynin.We consider the complex plane

*C*as a space filled by two different media, separated by the real axis*R*. Let*H*_{+}={*z*: Im*z*> 0} be the upper half-plane. For a planar body*E*, the general iceberg-type problem is to estimate characteristics of the invisible part,*E*_{_}=*E*\*H*_{+}, from characteristics of the whole body*E*and its visible part,*E*_{+}=*E*∩*H*_{+}. In this paper, we find the maximal draft of*E*as a function of the logarithmic capacity of*E*and the area of*E*_{+}.

**A Proof of Campbell's Subordination Conjecture**with R.W. Barnard.In the early 70's, D.M. Campbell published three papers on majorization-subordination results for locally univalent functions. In particular, he showed that if

*F*is linearly invariant of order α and if*f*is subordinate to*F*on {z : |z| < 1}, then*f '*is majorized by*F '*on { z : |z| < m(α)} where m(α) = α +1 - sqrt{α^{2}+2α}, provided α ≥ 1.65. He conjectured, in fact, that this result also held for 1.65 > α ≥ 1. We will review Campbell's proof and why the restriction α ≥ 1.65 arose in the proof. We will then affirmatively verify Campbell's conjecture.

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**The Verification of an Inequality**" with Roger W. Barnard.In a recent paper, we verifed a conjecture of Mej\'ia and Pommerenke that the extremal value for the Schwarzian derivative of a hyperbolically convex function is realized by a symmetric hyperbolic ``strip'' mapping. There were three major steps in the verification: first, a variational argument was given to reduce the problem to hyperbolic polygons bounded by at most two hyperbolic geodesics; second, a reduction was made to hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics; third, for hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics a computation was made, using properties of special functions, to find the maximal value of the Schwarzian derviative.

In between the second and third steps, an assertion was made that "using an extensive computational argument which considers several cases" the problem of computing the Schwarzian derivative for hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics could be reduced to computing the Schwarzian derivative for hyperbolic polygons bounded by exactly two symmetric hyperbolic geodesics under the assumption that the argument

*z*of the Schwarzian derviative satisfied the restriction 0 ≤*z*< 1. In this paper, we provide a verification for that assertion.

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**A Sharp Bound On The Schwarzian Derivatives Of Hyperbolically Convex Functions**" with Roger W. Barnard, Leah Cole and G. Brock Williams.We complete the determination of how far convex maps can distort discs in each of the three classical geometries. The euclidean case was settled by Nehari in 1976, and the spherical case by Mej\'ia and Pommerenke in 2000. We find sharp bounds on the Schwarzian derivatives of hyperbolically convex functions and thus complete the hyperbolic case. This problem was first posed by Ma and Minda in a series of papers in the 1980's. Mej\'ia and Pommerenke then produced partial results and a conjecture as to the extremal function. Their function maps onto a domain bounded by two proper geodesic sides, a "hyperbolic strip." Applying a generalization of the Julia variation and a critical Step Down Lemma, we show that there is an extremal function mapping onto a domain with at most two geodesic sides. We then verify using special function theory that among the remaining candidates, Mej\'ia and Pommerenke's two-sided function is in fact extremal. This correlates nicely with the euclidean and spherically convex cases in which the extremal is known to be a map onto a two-sided "strip."

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**Three Extremal Problems for Hyperbolically Convex Functions**" with R.W. Barnard and G.B. Williams.In this paper we apply a variational method to three extremal problems for hyperbolically convex functions posed by Ma and Minda and Pommerenke. We first consider the problem of extremizing Re

*f*(*z*)/*z*. We determine the minimal value and give a new proof of the maximal value previously determined by Ma and Minda. We also describe the geometry of the hyperbolically convex functions*f*(*z*)=α*z*+*a*_{2}*z*^{2}+*a*_{3}*z*^{3}+ . . . which maximize Re*a*_{3}.

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**A Variational Method For Hyperbolically Convex Functions**" with Roger W. Barnard and G. Lenny OrnasIn this paper we describe a variational method, based on Julia's formula for the Hadamard variation, for hyperbolically convex polygons. We use this variational method to prove a general theorem for solving extremal problems for hyperbolically convex functions. Special cases of this theorem provide independent proofs for controlling growth and distortion for hyperbolically convex functions.

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**Area, Width, And Logarithmic Capacity Of Convex Sets**," with R.W. Barnard and A.Y. Solynin.For a planar convex compact set

*E*, we describe the mutual range of its area, width, and logarithmic capacity. This result will follow from a more general theorem describing the mutual range of area, logarithmic capacity, and length of orthogonal projection onto a given axis of an arbitrary compact set, connected or not.

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**An Isoperimetric Inequality for Logarithmic Capacity**," with R.W. Barnard and A.Y. Solynin.We prove a sharp lower bound of the form cap

*E*≥ (1/2)diam*E*·Ψ(area*E*/((π/4) diam^{2}*E*)) for the logarithmic capacity of a compact connected planar set*E*in terms of its area and diameter. Our lower bound includes as special cases G. Faber's inequality cap*E*≥ (1/4)diam*E*and G. Pòlya's inequality cap*E*≥ (area*E*/π)^{1/2}. We give explicit formulations, functions of (1/2)diam*E*, for the extremal domains which we identify.

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**Creating a Refereed Electronic Mathematics Journal**," with G.A. Harris and S. Cooper.In the fall of 1996 we received National Science Foundation funding through the Dana Center at the University of Texas to develop a website dedicated to issues in the undergraduate mathematics preparation of schoolteachers, IUMPST. [http://www.k-12prep.math.ttu.edu]. The purpose of this paper is to share our experiences and lessons learned from the creation of IUMPST: The Journal. We will consider issues, both practical and philosophical, involving the journal's development and implementation.

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**A Survey of Applications of the Julia Variation**," with Roger W. Barnard and Carolynn Campbell.This is an introductory survey on applications of the Julia variation to problems in geometric function theory. A short exposition is given which develops a method for treating extremal problems over classes

*F*of analytic functions on the unit disk*D*for which appropriate subsets*F*can be constructed so that (i)_{n}*F*= [^{−}(∪)] and (ii) for each_{n}F_{n}*f*∈*F*a geometric constraint will hold that ∂_{n}*f*(*D*) will have at most*n*"sides". Applications of this method which have been made to problems in the literature are reviewed, e.g., Netanyahu's problems about the distortion theorems for starlike and convex functions constrained to contain a fixed disk; Goodman's problems about omitted values for classes of univalent functions; integral means estimates for derivatives of convex functions; maximization problems for functionals on linear fractional transforms of convex and starlike functions.

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**Inequalities for the Perimeter of an Ellipse**," with Roger W. Barnard and Lawrence Schovanec.

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**Zeros of Cesàro Sum Approximations**," with R.W. Barnard and W. Wheeler.Let

*D*denote the open unit disk and let*f*(*z*) = ∑_{n=0}^{∞}*a*_{n}*z*^{n}be analytic on*D*with positive monotone decreasing coefficients*a*_{n}. We answer several questions posed by J. Cima on the location of the zeros of polynomial approximants which he originally posed about outer functions. In particular, we show that the zeros of Cesàro approximants to*f*are well-behaved in the following sense: (1) if [(*a*_{n})/(*a*_{n+1})] → 1, and [(*a*_{0})/(*a*_{m})] ≤*am*^{b}, then ∂*D*is the only accumulation set for the zeros of the Cesàro sums of*f*; and (2) if*f*has a representation*f*(*z*) = ∑_{n=0}^{∞}*g*( [1/(*n*+*c*)] )*z*^{n}where*g*(*x*) = ∑_{n=0}^{∞}*b*_{n}*x*^{n}≢ 0,*b*_{n}≥ 0, then we give sufficient conditions so that the convex hull of the zeros of the Cesàro sums of*f*will contain*D*.

**A Monotonicity Property Involving**", with R.W. Barnard and K. Richards._{3}*F*_{2}and Comparisons of Classical Approximations of Ellipitcal Arc LengthConditions are determined under which

_{3}*F*_{2}(-*n*,*a*,*b*;*a*+*b*+2,ε-*n*+1;1) is a monotone function of*n*satisfying*a**b*·_{3}*F*_{2}(-*n*,*a*,*b*;*a*+*b*+2,ε-*n*+1;1) ≥*a**b*·_{2}*F*_{1}(*a*,*b*;*a*+*b*+2;1). Motivated by a conjecture of M. Vuorinen, the corollary that_{3}*F*_{2}(-*n*,-^{1}/_{2},-^{1}/_{2};1,ε-*n*+1;1) ≥ 4/π, for 1 > ε ≥^{1}/_{4}and*n*≥ 2, is used in the verification of a complete list of inequalities comparing the Maclaurin series coefficients of_{2}*F*_{1}with the coefficients of the known historical approximations of the arc length of an ellipse, for which maximum errors can then be established.

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**The Effects of Placement Examinations and Enforcing Prerequisites on Student Success in Entry-Level Mathematics Courses**", with R.M. Anderson.Prior to 1995, the placement criteria for entry-level mathematics courses at Texas Tech University were a combination of stipulated high school background requirements and SATM/ACTM score requirements. Advisors were allowed latitude to consider alternate factors for students with marginal scores. As a result, students were admitted into courses where they did not meet the formal prerequisites. There were various consequences of this process which led to dissatisfaction. Two major concerns were: student success rates in mathematics courses, especially in sequential courses such as calculus; and pedagogical issues related to non-homogenous student populations.

In Fall 1996, a mandatory, university-wide placement program was implemented for the entry-level mathematics courses with a requirement that the prerequisites be enforced. There were various immediate impacts on course population distributions with the most obvious being that our remediation population jumped from 629 (Fall 1994) to 1,239 (Fall 1996).

We now have data summaries from this first year 1996-97 in which the mandatory placement program was instituted. We also have contrasts with data generated from 1994-95. The data summaries show statistically significant differences for student success rates for various comparable population strata and a potential explanation for those areas in which no difference was detectable.

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**On the Muir-Ramanujan Approximation to the Arclength of an Ellipse**", with R.W. Barnard and K. Richards.In this paper we verify a conjecture of M. Vuorinen that the Muir-Ramanujan approximation is a lower approximation to the arc length of an ellipse. In particular, Vuorinen conjectured that

*f*(*x*) =_{2}*F*_{1}(^{1}/_{2},-^{1}/_{2};1;*x*)-((1+(1-*x*)^{3/4})/2)^{2/3}is positive for*x*∈ (0,1). The authors prove a much stronger result which says that the coefficients of*f*are nonnegative. As a key lemma, we show that_{3}*F*_{2}(-*n*,*a*,*b*;1+*a*+*b*,1+ε-*n*;1) > 0 when 0 <*ab*/(1+*a*+*b*) < ε < 1 for all positive integers*n*.

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**Cesàro Sum Approximations of Outer Functions**" with R.W. Barnard and J. Cima.It is well know that outer functions are zero-free on the unit disk. If an outer function,

*f*, is given as an infinite series and a finite (polynomial approximation is chosen, then it is desirable that the approximants retain the zero-free property of*f*. We observe for outer functions that the standard Taylor approximants do not, in general, retain the zero-free property - even when fairly restrictive conditions are placed on the permissible outer functions. We show, using methods of geometric function theory, that Cesàro sum approximants for outer functions which arise as the derivatives of bounded convex functions do inherit the desired zero-free property. We, also, find that a "cone-like" condition holds for the boundaries of the ranges of these approximants.

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**Gram Polynomials and the Kummer Function**" with R.W. Barnard, G. Dahlquist, L. Reichel and K.C. RichardsLet {φ

_{k}}^{n}_{k = 0},*n*<*m*, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form These polynomials are known as Gram polynomials. The present paper investigates the growth of |φ( *g*,*h*)_{d}: =1

*m**m*

∑

*j*= 1*g*(*x*_{j})*h*(*x*_{j}),*x*_{j}: = -1+(2*j*-1)/*m*._{k}(*x*)| as a function of*k*and*m*for fixed*x*∈ [-1,1]. We show that when*n*≤ 2.5*m*^{1/2}, the polynomials in the family {φ_{k}}^{n}_{k = 0}are of modest size on [-1,1], and they are therefore well suited for the approximation of functions on this interval. We also demonstrate that if the degree*k*is close to*m*, and*m*≥ 10, then φ_{k}(*x*) oscillates with large amplitude for values of*x*near the endpoints of [-1,1], and this behavior makes φ_{k}poorly suited for the approximation of functions on [-1,1]. We study the growth properties of |φ_{k}(*x*)| by deriving a second order differential equation, one solution of which exposes the growth. The connection between Gram polynomials and this solution to the differential equation suggested what became a long-standing conjectured inequality for the confluent hypergeometric function_{1}*F*_{1}, also known as Kummer's function, i.e., that_{1}*F*_{1}((1-*a*)/2,1,*t*^{2}) ≤_{1}*F*_{1}(1/2,1,*t*^{2}) for all*a*≥ 0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants. - "
**On a Coefficient Conjecture of Brannan**" with R.W. Barnard and W. WheelerIn 1972, D.A. Brannan conjectured that all of the odd coefficients,

*a*_{2n+1}, of the power series (1+*xz*)^{α}/(1-*z*) were dominated by those of the series (1+*z*)^{α}/(1-*z*) for the parameter range 0<α<1, after having shown that this was not true for the even coefficients. He verified the case when 2*n*+1 = 3. The case when 2*n*+1 = 5 was verified in the mid-eighties by J.G. Milcetich. In this paper, we verify the case when 2*n*+1 = 7 using classical Sturm sequence arguments and some computer algebra. - "
**A Short Proof of a Conjecture on the Integral Means of the Derivative of a Convex Function**" with R.W. BarnardThe authors obtained in a previous paper sharp estimates for the integral means [1/( 2 π)] ∫

_{0}^{2 π}|*f*′(*e*^{i θ}) |^{ -1}*d*θ for convex univalent*f*. In this paper, they devise a new, simpler proof of an analogous theorem for a more general class of functions. - "
**An Application from the Partial Sums of e**," with R.W. Barnard and R.S. Varga^{z}to a Problem in Several Complex VariablesLet

*B*_{r}^{n}= {*z*∈*C*^{n}: |*z*| <*r*}, where |·| is the Euclidean norm, and for*X*⊂*C*^{n}, let*HX*denote the closed convex hull of*X*in*C*^{n}. In 1990, Graham showed that if*f*is a normalized holomorphic map from*B*_{1}^{n}into*C*^{n}, and if*f*is either an open map or a polynomial map, then there is sharp, uniform constant*a*,*a*given by*ae*^{1+a}= 1, such that*Hf*(*B*_{1}^{n}) contains*B*_{a}^{n}. Graham posed the question to find, for normalized polynomial maps*f*of degree*m*, the best constant*a*_{m}so that*Hf*(*B*_{1}^{n}) contains*B*_{am}^{n}. We answer this question and obtain, for each*m*, the sharp constant*a*_{m}=*a*+*a*ln*m*

2(1+*a*)*m*+ *a*ln{ (2 π)^{1/2}(1+*a*)/*a*}

(1+*a*)*m*+ *o*(1

*m*), *m*→ ∞.We also note that this solution extends an old result of Pòlya and Szegö.

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**Sharp Bounds on the**" with R.W Barnard*H*Means of the Derivative of a Convex Function for_{p}*p*= -1We prove that if

*f*maps the unit disk*D*onto a convex domain with*f*(0) = 0, then1

2π⌠

⌡2π

01/| *f*′(*e*^{i θ})|*d*θ ≤ 1/min | *f*(*e*^{i θ}) |,with equality holding for each

*f*such that∂*f*(*D*) has the property that the left- and right-hand tangents at each point*w*∈ ∂*f*(*D*) are tangent to the circle {*w*: |*w*| = min|*f*(*e*^{i θ}) | }. This verifies the conjecture which we made in the early 1980's, which arose out of our work on omitted value problems for classes of univalent functions.This conjecture was announced at the Purdue Symposium on the Occasion of the Proof of the Bieberbach Conjecture in 1985 and published in the article Öpen Problems and Conjectures in Complex Analysis" (

*Computational Methods and Function Theory*, Lecture Notes in Math. #1435, 1990 pp. 1-26).The proof employs the Julia variation and properties of convexity and subordination.

Also, we obtain, using this integral means estimate, a sharp bound for the convex case of Brennan's conjecutre on the integral means with respect to area measure for the derivative of a univalent function.

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**Polynomials with Nonnegative Coefficients**" with R.W. Barnard, W. Dayawansa and D. WeinbergThe authors verify the conjecture that a conjugate pair of zeros can be factored from a polynomial with nonnegative coefficients so that the resulting polynomial still has nonnegative coefficients. The conjecture was originally posed by A. Rigler, S. Trimble and R. Varga arising out of their work on the Beauzamy-Enflo generalization of Jensen's inequality. The conjecture was also made independently by B. Conroy in connection with his work in number theory. A crucial and interesting lemma is proved which describes general coefficient-root relations for polynomials with nonnegative coefficients and for polynomials for which the case of equality holds in Descarte's Rule of Signs.

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**A Constructive Method for Numerically Computing Conformal Mappings for Gearlike Domains**"An algortihm and program are described for computing the conformal mapping function from the unit disk to a gearlike domain, i.e., a domain containing the origin whose boundary consists of arcs of circles centered at the origin and segments of radial lines emanating from the origin. The gearlike domain may be bounded or unbounded and may have slits. Both the forward and the inverse mapping are computable. Several examples are illustrated.

An application of the program is given which shows that a coefficient conjecture of R.W. Barnard for bounded univalent starlike functions fails.

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**Rounding Corners of Gearlike Domains and the Omitted Area Problem**" with R.W. BarnardA function

*g*analytic on the open unit disk*D*and vanishing only at the origin is said to be gearlike if*g*maps*D*to a domain whose boundary consists of arcs of circles centered at the origin and segments of rays emanating from the origin.

The authors discuss eachof the possible types of (boundary) corners the image domain of gearlike functions may have and give formulae for rounding or smoothing each of these possible corners, extending some early work of P. Henrici.

The omitted area problem, first posed by Goodman in 1949, is to determine for a normalized univalent analytic function*f*on*D*the maximum areas in*D*which can be omitted from the range of*f*. While Goodman gave some early bounds for the maximal omitted area, the problem has generally proved to be one of the difficult and long outstanding problems in geometric function theory. The authors apply the method of rounding corners to a specifically constructed gearlike function to produce an approximation for the extremal solution. - "
**Sums of Exponentials with Polynomial Coefficients**" with C.F. Martin and J. Miller - "
**Numerical Solution of Positive Sum Exponential Equations**" with C.F. Martin and J. MillerA homotopy continuation method is applied to solve an inverse interpolation problem for representing data by a positive sum of decaying exponentials. The homtopy method transforms the interpolation problem to a problem of determining the roots of a given polynomial. The relative effectiveness of the continuation method is constrasted with several other root-finding schemes.

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**Parameter Identification by Continuation Methods**" with C.F. Martin and J. MillerA method is discussed for calculating a Dirichlet polynomial with positive coefficients. The method employs continuation techniques used for finding the zeros of a mapping in

*n*-dimensional Euclidean space*R*.^{n} - "
**A Note on a Problem of Robinson**"Let

*S*be the usual class of univalent analytic functions on |*z*| < 1 normalized by*f*(0) = 0 and*f*′(0) = 1. Let*L*be the linear operator on*S*given by*Lf*= 1/2 (*zf*)′ and let*r*be the minimum radius of starlikeness of_{St}*Lf*for*f*in*S*. In 1947 R.M. Robinson initiated the study of properties of*L*acting on*S*when he showed that*r*> 0.38. Later, in 1975 R.W. Barnard gave an example which showed that_{St}*r*< 0.445. It is shown here, using a distortion theorem and Jenkin's region of variability for_{St}*zf*′(*z*)/*f*(*z*),*f*in*S*, that*r*> 0.435. Also, a simple example, a close-to-convex half-line mapping, is given which again shows_{St}*r*< 0.445._{St} - "
**A Product Theorem for**"*F*Classes and an Application_{p}For ℜ

*p*> 0 let*F*_{p}= {*f*:*f*(*z*) = ∫_{|x| = 1}(1 -*xz*)^{-p}*d*μ(*x*), |*z*| < 1 , μ a probability measure on |*x*| = 1 } and let*F*_{p}·*F*_{q}= {*fg*:*f*∈*F*_{p},*g*∈*F*_{q}}. Brickman, Hallenbeck MacGregor and Wilken proved a product theorem for the*F*_{p}classes; they showed that if*p*> 0,*q*> 0, then*F*_{p}·*F*_{q}is a subset of*F*_{p+q}. We give an (essentially complete) converse for the result of Brickman, et al., i.e., we show that if*F*_{p}·*F*_{q}is a subset of*F*_{p+q}, then*p*> 0,*q*> 0 or else*p*= 1 +*it*for some*t*real. As an intermediate consequence we disprove a conjecture about the extreme points of the closed convex hulls of the classes*Sp*(γ), 0 < |γ| < π/2, of γ- spirallike univalent functions, i.e., writing*m*= 1 +*e*^{-2i γ}, we show {*z*/(1 -*xz*)^{m}: |*x*| = 1 } is a proper subset of*EHSp*(γ), 0 < |γ| < π/2. - "
**New Support Points of***S*and Extreme Points of*HS*Let

*S*be the usual class of univalent analytic functions*f*on {*z*: |*z*| < 1 } normalized by*f*(*z*) =*z*+*a*_{2}*z*^{2}+ …. We prove that the functions*f*_{x, y}(*z*) =*z*- 1/2 (*x*+*y*)*z*^{2}

(1 -*yz*)^{2}, | *x*| = |*y*| = 1,*x*&ne*y*which are support points of

*C*, the subclass of*S*of close-to-convex functions, and extreme points of*HC*, are support points of*S*and extreme points of*HS*whenever 0 < |arg(-*x*/*y*)| ≤ π/4. We observe that the known bound of π/4 for the acute angle between the omitted arc of a support point of*S*and the radius vector is achieved by the functions*f*_{x, y}with |arg(-*x*/*y*)| = π/4. - "
**Generalized Bazilevic Functions**" with D.M. CampbellThis paper defines the generalized Bazilevic functions via the differential equation

1 + *z**f*″(*z*)

*f*′(*z*)+ (α+ *i*β-1)*z**f*′(*z*)

*f*(*z*)= α *z**g*′(*z*)

*g*(*z*)+ *z**h*′(*z*)

*h*(*z*)+ *i*βwhere α+

*i*β belongs to*C*\ {negative integers} and*g*,*h*are restricted to various function classes. The geometry of the solutions, their representation, the relation of their univalence to the domain of analyticity, and the motivation for considering the planar projection of the various representations of a generalized Bazilevic function are considered. The extremal problems max|*a*_{2}|, max|*a*_{3}| are solved. An explicit bound on the radius of Bazilevicness for*S*is obtained. A bounded univalent non-Bazilevic function which is a generalized Bazilevic function is constructed. Thomas' result that bounded*B*(α,0) functions satisfy |*a*_{n}| =*O*(*n*^{-1}) is generalized to classes of non-univalent functions. The paper closes with a conjecture on the analytic structure of a bounded univalent function whose coefficients satisfy |*a*_{n}| =*O*(*n*^{-1}).

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