On q-Calculus and Starlike Functions

Theorem 1 The function gðzÞ ¼ z þ az2 is in the class Sðf; aÞ, if and only if Re 1  jaj2ðf þ 1Þ  jajjfj 1  jaj2 ( ) [ a: ð13Þ

Lemma 1 Ruscheweyh and Sheil-Small (1973) If f 2 Sð1=2Þ and g 2 Sð1=2Þ [or if f 2 K and g 2 S , then fðzÞ  gðzÞFðzÞ fðzÞ  gðzÞ 2 cofFðDÞg; z 2 D; ð14Þ where F 2 H and cofFðDÞg denotes the closed convex hull of FðDÞ.

Theorem 2 If f is in the class Sð1=2Þ of starlike functions of order 1/2, then for all f, jfj  1, f is in the class S f; 1=ð1 þ jfj ð Þ of f-starlike functions of order 1=ð1 þ jfjÞ.

Corollary 1 If f is in the class Sð1=2Þ of starlike functions of order 1/2, then for all f, jfj  1, f is in the class Sðf; 1=2Þ of f-starlike functions of order 1/2.

Corollary 1 provides some examples of f-starlike func- tions of order 1/2, for example g 2 Sð1=2Þ ) g 2 Sðf; 1=2Þ; for all f; jfj  1: It is known that K  Sð1=2Þ; therefore, Corollary 1 leads to the following result.

Corollary 2 If f is in the class K of convex univalent functions, then for all f, jfj  1, f is in the class Sðf; 1=2Þ of f-starlike functions of order 1/2. We can look for the smallest a such that for all f, jfj  1, we have K  Sðf; aÞ. From

Corollary 2, we have 0\a  1=2 and it is known that for f ¼ 1 the order of a- starlikeness in the class of convex functions is 1/2, so 1/2 is the solution of this problem. However, we may consider this problem for a given f. Open problem. For given f, jfj  1, find the smallest a such that K  Sðf; aÞ: Recall here another definition of q-starlike functions of order a. Namely, making use of q-derivative (6), Argawal and Sahoo in Agrawal and Sahoo (2017) introduced the class S qðaÞ. A function f 2

Lemma 2 If f is in the class K of convex univalent func- tions, then we have Re ð1  fÞdffðzÞ f 0ðzÞ   [ 0; z; f 2 D; ð17Þ

Theorem 3 If f and h are in the class K of convex uni- valent functions, then we have Re hðzÞ  ð1  fÞzdffðzÞ hðzÞ  zf 0ðzÞ   [ 0; z; f 2 D: ð20Þ

Corollary 3 If f is in the class K of convex univalent functions, then we have Re ð1  fÞ R z 0 dffðtÞdt fðzÞ   [ 0; z; f 2 D: ð21Þ

Corollary 4 If f is in the class K of convex univalent functions, then we have Re ð1  fÞdffðzÞ f 0ðzÞ   [ 0; z; f 2 D: ð24Þ

Corollary 5 If f is in the class K of convex univalent functions, then we have Re ð1  fÞ R z 0 tdffðtÞdt R z 0 tf 0ðtÞdt ( ) [ 0; z; f 2 D: ð26Þ

Corollary 6 If f is in the class K of convex univalent functions, then we have Re ð1  fÞ R z 0 dfðtdffðtÞÞdt R z 0 dfðtf 0ðtÞÞdt ( ) [ 0; z; f 2 D: ð28Þ

Definition 1 [11] Let f 2 A. For given f, jfj  1, we say that f is in the class Sðf; aÞ of f-starlike functions of order a, 0  a\1, if Re zdffðzÞ fðzÞ   [ a; z 2 D;