Evelyn Silvia

Some of Professor Silvia's most recent papers concern classes of functions that are related to an interesting subclass, D, of convex functions that was originally considered by St. Ruscheweyh. Silvia and Silverman [3] determined the largest dilation factor d(a) such that f(d(a)z)/d(a) is guaranteed to be in class D, assuming that f itself is either convex of order a or satisfies Re(f'(z)) > a. The thrust of this result is that, if a function satisfies strong geometric conditions in a small disk, then its analytic continuation to a larger disk will still satisfy interesting convexity conditions. Other related works look at integral and convolution (Hadamard product) characterizations of classes related to D.

Professor Silvia [5] has recently completed a study of convex null sequences. A decreasing sequence of real numbers is called convex null if it converges to zero and if each term is less than or equal to the average of the two adjacent terms. A classical result of Fejer relates these sequences to coefficients of functions with positive real part. Professor Silvia's paper on these sequences gave a class of functions for which preservation properties could be proved using facts about convex null sequences in addition to offering a new general theoretical construct along with meaningful applications.

Professor Silvia is also seriously involved in mathematics education. She has emphasized curriculum development at all levels as well as teacher training and teacher enhancement. She was co-director of UCDavis' first MAT in mathematics program, 1973-79. This program introduced students to innovative teaching methods involving large-group {Socratic teaching}. Recently, she designed a problem-solving curriculum in connection with an Eisenhower Grant for a three year project in West Sacramento. She is currently the principal investigator for the Northern California Mathematics Project. At the university level, she is a co-director (with Dr. Carol Hom) of the UC Davis Calculus Revitalization Project. Professor Silvia has also written a preliminary version of an introductory graduate textbook/workbook in complex analysis that is in use in the graduate program at Davis. She is currently co-authoring Introduction to Abstract Mathematics--A Working Excursion Through Analysis with Professor Emeritus Doyle Cutler.

Selected publications

[2] Property preserving operators, Bull. Austral. Math. Soc. 39 (1989), 397-404.

[3] Radii properties for subclasses of convex functions (with H. Silverman), J. Math. Anal. Appl. 194 (1995), 428-436.

[4] Personalized teaching in large classes (with C. L. Hom), Primus, 6 (1996) 325-336.

[5] Convex null operators. Math. Japon. 47 (1998), 311-317.

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