Hindawi Abstract and Applied Analysis Volume 2019, Article ID 6947020, 6 pages https://doi.org/10.1155/2019/6947020 Research Article Certain Subclasses of Bi-Close-to-Convex Functions Associated with Quasi-Subordination Gurmeet Singh ,1 Gurcharanjit Singh,2 and Gagandeep Singh 3 Principal, Patel Memorial National College, Rajpura, Punjab, India Research Scholar, Department of Mathematics, Punjabi University, Patiala, Punjab, India Department of Mathematics, Majha College for Women, Tarn Taran Sahib, Punjab, India Correspondence should be addressed to Gagandeep Singh; [email protected] Received 13 January 2019; Accepted 7 March 2019; Published 1 April 2019 Academic Editor: Luigi Rodino Copyright © 2019 Gurmeet Singh et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the present investigation, we introduce certain new subclasses of the class of biunivalent functions in the open unit disc �={�: |�| < 1} defned by quasi-subordination. We obtained estimates on the initial coefcients |�2| and |�3| for the functions in these subclasses. Te results present in this paper would generalize and improve those in related works of several earlier authors. ��� (�) 1. Introduction and Preliminaries �={�:�∈�, ( )>0,�(�) ∈�∗;� Re � (�) Let � be the class of functions of the form ∞ � � (�) =�+∑ ��� (1) ∈�}, the class of close-to-convex functions. �=2 which are analytic in the open unit disc �={�:|�|<1}. � { (��� (�)) Further, let � be the class of functions �(�) ∈ � and univalent �1 = {�:�∈�,Re ( )>0,ℎ(�) ∈�;� in �. ℎ� (�) { By �, we denote the class of bounded or Schwarz func- tions �(�) satisfying �(0) = 0 and |�(�)| ≤ 1 which are � } analytic in the unit disc and of the form ∈�} , the class of quasi-convex functions. ∞ � } � (�) = ∑ ��� ,�∈�. (2) �=1 (3) Firstly, it is necessary to recall some fundamental defni- Te functions in the class � are invertible but their inverse tions to acquaint with the main content: function may not be defned on the entire unit disc �.Te ��� (�) � �∗ ={�:�∈�, ( )>0;�∈�}, Koebe-one-quarter theorem [1] ensures that the image of Re � (�) under every function �∈�contains a disc of radius 1/4. −1 Tus every univalent function � has an inverse � , defned the class of starlike functions. by � { (��� (�)) } −1 �= �:�∈�, ( )>0;�∈� , � (� (�))=� (�∈�) , { Re �� (�) } { } (4) −1 1 �(� (�))=�(|�| <�0 (�) : �0 (�) ≥ ) the class of convex functions. 4 2 Abstract and Applied Analysis 2 where � (�) =�0 +�1�+�2� +... −1 (10) � (�) =� (�) (|� (�)| ≤1,�∈�) . 2 2 3 =�−�2� +(2�2 −�3)� (5) Definition . For 0≤�≤1,afunction�∈Σgiven by (1) is said to be in the class �Σ(�, �, �) if there exists a bistarlike −(5�3 −5�� +�)�4 +... ∞ � 2 2 3 4 function �(�) = � + ∑�=2 ��� such that � Afunction�∈�is said to be biunivalent in � if both � and � � −1 1 �� (�) (�� (�)) � are univalent in U. [(1−�) +� −1] �∈� � � (�) �� (�) Accordingly, a function is said to be bistarlike, [ ] (11) biconvex, bi-close-to-convex, or bi-quasi-convex if both � −1 and � are starlike, convex, close-to-convex, or quasi-convex ≺� (� (�) −1), respectively. � � Let Σ denote the class of biunivalent functions in � given 1 �ℎ� (�) (�ℎ (�)) [(1−�) +� −1] by (1). Examples of functions in the class Σ are � � (�) �� (�) � [ ] (12) , 1−� ≺� (� (�) −1), − log (1−�) , (6) −1 −1 where ℎ=� , �=� ,and�, � ∈ �. 1 1+� �=0 � (�, �, �) � (�, �) ( ), For ,theclass Σ reduces to Σ , 2 log 1−� the class of bi-close-to-convex functions of complex order � defned by quasi-subordination. and so on. However, the familiar Koebe function �(�) = �/(1 − �)2 Σ is not a member of . Definition . For 0≤�≤1,afunction�∈Σgiven by (1) is Let � and � be two analytic functions in �.Ten� is said 1 said to be in the class �Σ(�, �, �) if there exists a biconvex to be subordinate to � (symbolically �≺�)ifthereexistsa ∞ � function �(�) = � + ∑ ��� and satisfy the following bounded function �(�) ∈ � such that �(�) = �(�(�)).Tis �=2 conditions: result is known as principle of subordination. � � Robertson [2] introduced the concept of quasi- 1 ��� (�) (�� (�)) subordination in 1970. For two analytic functions � [(1−�) +� −1] � � (�) �� (�) and �,thefunction� is said to be quasi-subordinate to � [ ] (13) (symbolically �≺� �) if there exist analytic functions � and � with |�(�)| ≤ 1, �(0) = 0 and |�(�)| < 1 such that ≺� (� (�) −1), � (�) � � ≺�(�) , (7) 1 �V� (�) (�V (�)) � (�) [(1−�) +� −1] � � (�) �� (�) or equivalently [ ] (14) � (�) =�(�) � (� (�)) . (8) ≺� (� (�) −1), �(�) = 1 �(�) = �(�(�)) �(�) ≺ −1 −1 Particularly if ,then ,sothat where V =� , �=� ,and�, � ∈ �. �(�) in �. So it is obvious that the quasi-subordination is a 1 It is interesting to note that, for �=0, � (0, �, �) is the generalization of the usual subordination. Te work on quasi- Σ subclass of bi-close-to-convex functions of complex order � subordination is quite extensive which includes some recent 1 defned by quasi-subordination. Also for �=1, �Σ(1, �, �) investigations [3–6]. � Lewin [7] investigated the class Σ of biunivalent functions is the class of bi-quasi-convex functions of complex order and obtained the bound for the second coefcient. Brannan defned by quasi-subordination. and Taha [8] considered certain subclasses of biunivalent For deriving our main results, we need the following functions, similar to the familiar subclasses of univalent lemmas. functions consisting of strongly starlike, starlike, and convex functions. Tey introduced bistarlike functions and biconvex Lemma 3 (see [12]). If �∈�is family of all functions � functions and obtained estimates on the initial coefcients. analytic in � for which Re[�(�)] > 0 and have the form Also the subclasses of bi-close-to-convex functions were 2 �(�) = 1 + �1�+�2� + ... for �∈�,then|��|≤2for studied by various authors [9–11]. each �. Motivated by earlier work on bi-close-to-convex and ∞ � quasi-subordination, we defne the following subclasses. Lemma 4 (see [13]). If �(�) = � + ∑ ��� is a starlike �(�) � �(0) = 1 �=2 Also it is assumed that is analytic in with function, then and let � � � 2� 2 + ��3 −�2 � ≤1. (15) � (�) =1+�1�+�2� +... (�1 ∈� ), (9) � � Abstract and Applied Analysis 3 ∞ � Lemma 5 (see [13]). If �(�) = � + ∑�=2 ��� is a convex Using (21) and (22) in (19) and (20), respectively, it yields function, then � � 1 ��� (�) (�� (�)) � � 1 [(1−�) +� −1] �� −�2� ≤ . � � (�) �� (�) � 3 2 � (16) 3 [ ] (23) � (�) −1 =�(�) [� ( )−1], Along with the above lemmas, the following well known results � (�) +1 are very useful to derive our main results. ∞ � � Let �(�) = �+∑ ��� be an analytic function in � of the � �=2 1 �ℎ� (�) (�ℎ (�)) form (), then |��|≤�,if�(�) is starlike and |��|≤1,if�(�) is [(1−�) +� −1] � convex. � � (�) � (�) [ ] (24) � (�) −1 =�(�) [� ( )−1]. 2. Coefficient Bounds for the Function Class � (�) +1 �Σ(�,�,�) But Teorem 6. If �∈�Σ(�, �, �),then � � 1 ��� (�) (�� (�)) 1 � � [ ] � � 1 �� �� � (1−�) +� −1 = [(1+�) �� � ≤ . [ [� 0 � 1 +2], � � (�) �� (�) � � 2� min 2 1+� [ ] [ (17) (25) � � � � � � ⋅(2� −�)�+((1+2�) (3� −�) 4 (1+4�) 2 (1+3�) �� �� � �� �� (� + �� −� �) 2 2 3 3 √ + � 0 � 1 + � 0 � 1 � 2 1� ] , 3 (1+2�) 3 (1+�)(1+2�) 3 (1+2�) 2 2 ] + (1+3�) (�2 −2�2�2)) � +...], 2 � � (4� +5�+2) � �2 1 � �� � ≤ �� �� �2 + � � � 3� 2 2 � 0 � 1 1 �ℎ (�) (�ℎ (�)) 1 (1+�) (1+2�) 3 [(1−�) +� −1] = [(1 � � � � � � � � � � � � (�) �� (�) � 4 �� �� � +(3�� �� �� −� � + �� �� � ) 2 �� �� � [ ] + � 0 � 1 � 0 � � 1 2� � 1 � 1 + � 0 � 1 3 (1+2�) (1+2�) (18) +�) (�2 −2�2)� (26) 2 2 � �2 � � � � (3� +3�+1)�1 ��� �� �� (� + �� −� �) ⋅ √ + � 0 � 1 � 1 2� . +((1+2�) [2 (3�2 −�2)−(3�−�)] (1+2�)2 (1+�)2 (1+2�) 2 2 3 3 2 2 + (1+3�) (�2 −2�2�2)) � +...]. Proof. As �∈�Σ(�, �, �), so by Defnition 1 and using the concept of quasi-subordination, there exist Schwarz Again using (9) and (10) in (21) and (22), respectively, we �(�) �(�) �(�) functions and and analytic function such that get � � � (�) −1 1 1 ��� (�) (�� (�)) � (�) [� ( )−1]= � � � � [(1−�) +� −1] � (�) +1 2 0 1 1 � � (�) �� (�) [ ] (19) 1 1 �2 � � �2 +[ � � � + � � (� − 1 )+ 0 2 1 ]�2 (27) =�(�) (� (� (�) −1) , 2 1 1 1 2 0 1 2 2 4 � � 1 �ℎ� (�) (�ℎ (�)) +..., [(1−�) +� −1] � � (�) �� (�) (20) � (�) −1 1 [ ] � (�) [� ( )−1]= � � � � � (�) +1 2 0 1 1 =�(�) (� (� (�)) −1) 2 2 1 1 �1 �0�2�1 (28) 2 2 +[ �1�1�1 + �0�1 (�2 − )+ ] where �(�) = 1+�1�+�2� +...and �(�) = 1+�1�+�2� +.... 2 2 2 4 Defne the functions �(�) and �(�) by ⋅�2 +... 2 � (�) −1 1 �1 2 � (�) = = [�1�+(�2 − )� +...], (21) Using (25) and (27) in (23) and equating the coefcients � (�) +1 2 2 2 of � and � ,weget � (�) −1 �2 1 1 2 (1+�) 1 � (�) = = [�1�+(�2 − )� +...]. (22) (2� −�)= � � � , � (�) +1 2 2 � 2 2 2 0 1 1 (29) 4 Abstract and Applied Analysis 2 (1+2�) (3�3 −�3)+(1+3�) (�2 −2�2�2) Now subtracting (32) from (30), we obtain � 2 (30) �3 −�2 2 2 � = +� 1 1 � � � 3 2 = � � � + � � (� − 1 )+ 0 2 �2. 3 2 1 1 1 2 0 1 2 2 4 1 (40) � � (� −� )+� � (� −� ) + 1 1 1 1 0 1 2 2 �. Again using (26) and (28) in (24) and equating the coefcients 12 (1+2�) 2 of � and � ,weget Applying triangle inequality and using Lemma 3 in (40), it (1+�) 1 (� −2�)= � � � , yields � 2 2 2 0 1 1 (31) � � �� −�2� � � � � 2 2 2 � � � 3 2 � � �2 (�� �� + �� ��)� (1+2�) [2 (3� −� )−(3�−�)] + (1+3�) (� −2�� ) �� � ≤ � � + �� � + � 1 � � 0 � 1 . (41) 2 2 3 3 2 2 2 � 3� 3 � 2� 3 (1+2�) � (32) 1 1 �2 � � Again adding (30) and (32) and applying triangle inequality, = � � � + � � (� − 1 )+ 0 2 �2. 2 1 1 1 2 0 1 2 2 4 1 we get From (29) and