RESEARCH ARTICLE The Radius of Convexity of Partial Sums of Convex Functions in One Direction Sudhananda Maharana 1 • Jugal K. Prajapat 1 • Hari M. Srivastava 2 Received: 5 March 2016 / Revised: 8 April 2016 / Accepted: 30 January 2017 Ó The National Academy of Sciences, India 2017 Abstract In this paper, we show that every section of function in a family of convex function in one direction in the open unit disk are convex in D 1=2 ¼fz 2 C : jzj\1=2g. The radius 1/2 is best possible. Keywords Analytic  Univalent  Close-to-convex  Starlike  Convex functions  Convex function in one direction  Radius of convexity  Hurwitz-Lerch zeta function Mathematics Subject Classification 30C45 1 Introduction and Statement of Main Result 1.1 Introduction Let for q [ 0, D q :¼fz 2 C : jzj\qg and D :¼ D 1 be the open unit disk. Let A denote the class of all analytic functions in D with the normalization f ð0Þ¼ 0 ¼ f 0 ð0Þ 1 and having the form f ðzÞ¼ z þ X 1 n¼2 a n z n ; z 2 D: ð1Þ Also we denote by S, the class of functions in A that are also univalent in D. A function f 2S is called starlike, if every line segment joining the origin to w 2 f ðDÞ lies completely inside f ðDÞ. The class of starlike functions in D is denoted by S  .A domain X is called convex in the direction u ð0  u\pÞ, if every line parallel to the line through 0 and e iu has a connected (or empty) intersection with X. A function f of the form (1) known to be convex in the direction u ð0  u\pÞ, if it maps D to the domain convex in direction u. Umezawa [1] studied that, if function f is locally uni- valent of the form (1) satisfying the relation a [ R 1 þ zf 00 ðzÞ f 0 ðzÞ   [  a 2a  3 ; ð2Þ where a is a real number not less than 3/2, then f is analytic and univalent in D. Moreover, f maps every D q , for every q\1 into a curve which is convex in one direction. Several special cases of inequality (2) can be drawn by allowing different values of a  3=2. In particular, if we allow a to approach 1 and 3/2 in (2), then we obtain the following subclasses of S F¼ f 2S : R 1 þ zf 00 ðzÞ f 0 ðzÞ   [  1 2 ; z 2 D   ; ð3Þ G¼ f 2S : R 1 þ zf 00 ðzÞ f 0 ðzÞ   \ 3 2 ; z 2 D   ; ð4Þ respectively. Observe that, the inequality appeared in (3) is a consequence of Kaplan characterization [2, p. 48, The- orem 2.18], hence functions in F are also close-to-convex in D. Ozaki [3] proved that functions in G are univalent in D, also Singh and Singh [4, Theorem 6] proved that functions in G are starlike in D. & Jugal K. Prajapat jkprajapat@curaj.ac.in Sudhananda Maharana snmmath@gmail.com Hari M. Srivastava harimsri@math.uvic.ca 1 Central University of Rajasthan, Dist-Ajmer, Bandarsindri, Kishangarh, Rajasthan 305817, India 2 University of Victoria, Victoria, BC V8W 3R4, Canada 123 Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. DOI 10.1007/s40010-017-0348-7