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Extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a two-parameter collection of kernel functions integrated against measures on the torus. For specific choices of the parameters many families from classical geometric function theory are included. These families include the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others. The main result introduces a surprising new class of extreme points.

We present a short geometric proof that the difference equation admits an invariant which is continuous and which separates solutions as efficiently as possible. The existence of invariants for a wide class of equations, including this one, has been recently established (see, R. Abus-Saris and Q. Al-Hassan, 2005 On globally attracting two-cycles of second order difference equations, Journal of Difference Equations and Applications 11, 1295–1303). This note presents an alternative and conceptually simpler approach.

The support points are obtained for the class of those analytic functions starlike of order α, m -fold symmetric that are real on (− 1, 1). We also determine the set of support points for the m -fold symmetric analytic functions which are starlike of order α.

Let D be a bounded domain in the complex plane C. Let H ∞ ( D ) {H^\infty }(D) denote the usual Banach algebra of bounded analytic functions on D. The Corona Conjecture asserts that D is weak ∗ \text {weak}^\ast dense in the space M ( D ) \mathfrak {M}(D) of maximal ideals of H ∞ ( D ) {H^\infty }(D) . In [2] Carleson proved that the unit disk Δ 0 {\Delta _0} is dense in M ( Δ 0 ) \mathfrak {M}({\Delta _0}) . In [7] Stout extended Carleson’s result to finitely connected domains. In [4] Gamelin showed that the problem is local. In [1] Behrens reduced the problem to very special types of infinitely connected domains and established the conjecture for a large class of such domains. In this paper we extract some of the crucial ingredients of Behrens’ methods and extend his results to a broader class of infinitely connected domains.

A special case of the two predator one prey model analyzed by Hsu, Hubbell and Waltman [Ecological Monographs, 48 (1978), pp. 337–349], [SIAM I. Appl. Math. 35 (1978), pp. 617–625] is discussed. Extending the techniques used there, a rigorous and complete analysis of the case in which the predators are most similar is presented. The analysis confirms the conclusions reached by Hsu, Hubbell and Waltman through computer calculations.

The stereospecificity of hydrogen transfer from NADPH to the appropriate carbonyl substrate catalyzed by ketopantoic acid and ketopantoyl acid and ketopantoyl lactone reductases of yeast (Saccharomyces cerevisiae) and Escherichia coli has been determined. Yeast and E. coli ketopantoic acid reductases are B-specific enzymes which transfer hydrogen from [4B-3H]-NADPH to ketopantoic acid to form [3H]pantoic acid. In contrast to the usual observations on the stereospecificity of functionally similar dehydrogenases from different species, yeast and E. coli ketopantoyl lactone reductases exhibit opposite stereospecificities. Both of two forms of yeast ketopantoyl lactone reductases are A-specific enzymes which form [3H]pantoyl lactone from ketopantoyl lactone and [4A-3H]NADPH, whereas, two forms of E. coli ketopantoyl lactone reductases are B-specific enzymes.

Let S denote the functions that are analytic and univalent in the open unit disk and satisfy f ( 0 ) = 0 f(0) = 0 and f ′ ( 0 ) = 1 f’(0) = 1 . Also, let K, St, S R {S_R} , and C be the subfamilies of S consisting of convex, starlike, real, and close-to-convex mappings, respectively. The closed convex hull of each of these four families is determined as well as the extreme points for each. Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets. The extreme points for each family are particularly simple; for example, the Koebe functions f ( z ) = z / ( 1 − x z ) 2 , | x | = 1 f(z) = z/{(1 - xz)^2},|x| = 1 , are the extreme points of cl co St. These results are applied to discuss linear extremal problems over each of the four families. A typical result is the following: Let J be a “nontrivial” continuous linear functional on the functions analytic in the unit disk. The only functions in St. that satisfy Re J ( f ) = max { Re J ( g ) : g ∈ S t } \operatorname {Re} \,J(f) = \max \;\{ \operatorname {Re} \;J(g):g \in St\} are Koebe functions and there are only a finite number of them.

notes the Banach algebra of all complex valued continuous functions on X with the supremum norm. A subalgebra A contained in C(X) is called a function algebra on X if it satisfies the following conditions. (i) A separates points on X. (ii) The constant functions are in A, i.e., 1 belongs to A. (iii) A is closed in C(X). For a function algebra A the space of maximal ideals is denoted by MA and the Silov boundary by SA. Both SA and MA carry natural compact Hausdorff topologies and SA and X can be imbedded in MA so that, identifying SA and X with their images in MA, we have SACXCMA. Moreover A, via restriction, is a function algebra on SA and, via the Gelfand representation, extends to a function algebra on MA. SA and MA are respectively the "smallest" and "largest" compact Hausdorff spaces on which A can be realized as a function algebra.

The stereospecificity of hydrogen transfer from NADPH to the appropriate carbonyl substrate catalyzed by ketopantoic acid and ketopantoyl acid and ketopantoyl lactone reductases of yeast (Saccharomyces cerevisiae) and Escherichia coli has been determined. Yeast and E. coli ketopantoic acid reductases are B-specific enzymes which transfer hydrogen from [4B-3H]-NADPH to ketopantoic acid to form [3H]pantoic acid. In contrast to the usual observations on the stereospecificity of functionally similar dehydrogenases from different species, yeast and E. coli ketopantoyl lactone reductases exhibit opposite stereospecificities. Both of two forms of yeast ketopantoyl lactone reductases are A-specific enzymes which form [3H]pantoyl lactone from ketopantoyl lactone and [4A-3H]NADPH, whereas, two forms of E. coli ketopantoyl lactone reductases are B-specific enzymes.