TY - JOUR

T1 - Notes on Convex Functions of Order (Formula presented.)

AU - Sugawa, Toshiyuki

AU - Wang, Li Mei

N1 - Funding Information:
The present research was supported by National Natural Science Foundation of China (No. 11326080) and JSPS Grant-in-Aid for Scientific Research (B) 22340025.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Marx and Strohhäcker showed in 1933 that f(z) / z is subordinate to (Formula presented.) for a normalized convex function f on the unit disk (Formula presented.) In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to (Formula presented.) if f is convex of order (Formula presented.) for (Formula presented.) and conjectured that this is true also for (Formula presented.) Here, (Formula presented.) is the standard extremal function in the class of normalized convex functions of order (Formula presented.) and (Formula presented.) We prove the conjecture and study geometric properties of convex functions of order (Formula presented.) In particular, we prove that (Formula presented.) is starlike whenever both f and g are convex of order 3 / 5.

AB - Marx and Strohhäcker showed in 1933 that f(z) / z is subordinate to (Formula presented.) for a normalized convex function f on the unit disk (Formula presented.) In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to (Formula presented.) if f is convex of order (Formula presented.) for (Formula presented.) and conjectured that this is true also for (Formula presented.) Here, (Formula presented.) is the standard extremal function in the class of normalized convex functions of order (Formula presented.) and (Formula presented.) We prove the conjecture and study geometric properties of convex functions of order (Formula presented.) In particular, we prove that (Formula presented.) is starlike whenever both f and g are convex of order 3 / 5.

KW - Convex functions of order (Formula presented.)

KW - Hypergeometric function

KW - Subordination

UR - http://www.scopus.com/inward/record.url?scp=84958543796&partnerID=8YFLogxK

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U2 - 10.1007/s40315-015-0122-2

DO - 10.1007/s40315-015-0122-2

M3 - Article

AN - SCOPUS:84958543796

SN - 1617-9447

VL - 16

SP - 79

EP - 92

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

IS - 1

ER -