AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2019/6947020 6947020 Research Article Certain Subclasses of Bi-Close-to-Convex Functions Associated with Quasi-Subordination http://orcid.org/0000-0003-0230-0990 Singh Gurmeet 1 Singh Gurcharanjit 2 http://orcid.org/0000-0001-5238-3603 Singh Gagandeep 3 Rodino Luigi 1 Principal Patel Memorial National College Rajpura Punjab India 2 Research Scholar Department of Mathematics Punjabi University Patiala Punjab India punjabiuniversity.ac.in 3 Department of Mathematics Majha College for Women Tarn Taran Sahib Punjab India 2019 142019 2019 13 01 2019 07 03 2019 142019 2019 Copyright © 2019 Gurmeet Singh et al. This 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 U = z : z < 1 defined by quasi-subordination. We obtained estimates on the initial coefficients a 2 and a 3 for the functions in these subclasses. The results present in this paper would generalize and improve those in related works of several earlier authors.

1. Introduction and Preliminaries

Let A be the class of functions of the form (1) f z = z + k = 2 a k z k which are analytic in the open unit disc U = z : z < 1 . Further, let S be the class of functions f ( z ) A and univalent in U .

By B , we denote the class of bounded or Schwarz functions w ( z ) satisfying w ( 0 ) = 0 and w z 1 which are analytic in the unit disc U and of the form (2) w z = n = 1 c n z n , z U .

Firstly, it is necessary to recall some fundamental definitions to acquaint with the main content: (3) S = f : f A , R e z f z f z > 0 ; z U ,   the  class  of  starlike  functions . K = f : f A , R e z f z f z > 0 ; z U ,   the  class  of  convex  functions . C = f : f A , R e z f z g z > 0 , g z S ; z U ,   the  class  of  close-to-convex  functions . C 1 = f : f A , R e z f z h z > 0 , h z K ; z U ,   the  class  of  quasi-convex  functions .

The functions in the class S are invertible but their inverse function may not be defined on the entire unit disc U . The Koebe-one-quarter theorem  ensures that the image of U under every function f S contains a disc of radius 1 / 4 . Thus every univalent function f has an inverse f - 1 , defined by (4) f - 1 f z = z z U , f f - 1 w = w w < r 0 f : r 0 f 1 4 where (5) g w = f - 1 w = w - a 2 w 2 + 2 a 2 2 - a 3 w 3 - 5 a 2 3 - 5 a 2 a 3 + a 4 w 4 + A function f A is said to be biunivalent in U if both f and f - 1 are univalent in U.

Accordingly, a function f A is said to be bistarlike, biconvex, bi-close-to-convex, or bi-quasi-convex if both f and f - 1 are starlike, convex, close-to-convex, or quasi-convex respectively.

Let Σ denote the class of biunivalent functions in U given by (1). Examples of functions in the class Σ are (6) z 1 - z , - log 1 - z , 1 2 log 1 + z 1 - z , and so on. However, the familiar Koebe function f ( z ) = z / ( 1 - z ) 2 is not a member of Σ .

Let f and g be two analytic functions in U . Then f is said to be subordinate to g (symbolically f g ) if there exists a bounded function u ( z ) B such that f ( z ) = g ( u ( z ) ) . This result is known as principle of subordination.

Robertson  introduced the concept of quasi-subordination in 1970. For two analytic functions f and ϕ , the function f is said to be quasi-subordinate to ϕ (symbolically f q ϕ ) if there exist analytic functions k and ω with k z 1 , ω ( 0 ) = 0 and ω z < 1 such that (7) f z k z ϕ z , or equivalently (8) f z = k z ϕ ω z . Particularly if k ( z ) = 1 , then f ( z ) = ϕ ( ω ( z ) ) , so that f ( z ) ϕ ( z ) in U . So it is obvious that the quasi-subordination is a generalization of the usual subordination. The work on quasi-subordination is quite extensive which includes some recent investigations .

Lewin  investigated the class Σ of biunivalent functions and obtained the bound for the second coefficient. Brannan and Taha  considered certain subclasses of biunivalent functions, similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions. They introduced bistarlike functions and biconvex functions and obtained estimates on the initial coefficients. Also the subclasses of bi-close-to-convex functions were studied by various authors .

Motivated by earlier work on bi-close-to-convex and quasi-subordination, we define the following subclasses.

Also it is assumed that ϕ ( z ) is analytic in U with ϕ ( 0 ) = 1 and let (9) ϕ z = 1 + B 1 z + B 2 z 2 + B 1 R + , (10) k z = A 0 + A 1 z + A 2 z 2 + k z 1 , z U .

Definition 1.

For 0 α 1 , a function f Σ given by (1) is said to be in the class C Σ ( α , γ , ϕ ) if there exists a bistarlike function g ( z ) = z + k = 2 b k z k such that (11) 1 γ 1 - α z f z g z + α z f z g z - 1 q ϕ z - 1 , (12) 1 γ 1 - α w h w j w + α w h w j w - 1 q ϕ w - 1 , where h = f - 1 , j = g - 1 , and z , w U .

For α = 0 , the class C Σ ( α , γ , ϕ ) reduces to C Σ ( γ , ϕ ) , the class of bi-close-to-convex functions of complex order γ defined by quasi-subordination.

Definition 2.

For 0 α 1 , a function f Σ given by (1) is said to be in the class C Σ 1 ( α , γ , ψ ) if there exists a biconvex function s ( z ) = z + k = 2 d k z k and satisfy the following conditions: (13) 1 γ 1 - α z f z s z + α z f z s z - 1 q ψ z - 1 , (14) 1 γ 1 - α w v w t w + α w v w t w - 1 q ψ w - 1 , where v = f - 1 , t = s - 1 , and z , w U .

It is interesting to note that, for α = 0 , C Σ 1 ( 0 , γ , ψ ) is the subclass of bi-close-to-convex functions of complex order γ defined by quasi-subordination. Also for α = 1 , C Σ 1 ( 1 , γ , ψ ) is the class of bi-quasi-convex functions of complex order γ defined by quasi-subordination.

For deriving our main results, we need the following lemmas.

Lemma 3 (see [<xref ref-type="bibr" rid="B8">12</xref>]).

If p P is family of all functions p analytic in U for which R e p z > 0 and have the form p ( z ) = 1 + p 1 z + p 2 z 2 + for z U , then p n 2 for each n .

Lemma 4 (see [<xref ref-type="bibr" rid="B4">13</xref>]).

If g ( z ) = z + k = 2 b k z k is a starlike function, then (15) b 3 - b 2 2 1 .

Lemma 5 (see [<xref ref-type="bibr" rid="B4">13</xref>]).

If g ( z ) = z + k = 2 b k z k is a convex function, then (16) b 3 - b 2 2 1 3 . Along with the above lemmas, the following well known results are very useful to derive our main results.

Let g ( z ) = z + k = 2 b k z k be an analytic function in A of the form (1), then b n n , if g ( z ) is starlike and b n 1 , if g ( z ) is convex.

2. Coefficient Bounds for the Function Class <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M112"> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi></mml:mrow> <mml:mrow> <mml:mi>Σ</mml:mi></mml:mrow> </mml:msub> <mml:mo mathvariant="bold">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo mathvariant="bold">,</mml:mo> <mml:mi>γ</mml:mi> <mml:mo mathvariant="bold">,</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula> Theorem 6.

If f C Σ ( α , γ , ϕ ) , then (17) a 2 m i n . 1 2 A 0 γ B 1 1 + α + 2 , 4 1 + 4 α 3 1 + 2 α + 2 1 + 3 α A 0 γ B 1 3 1 + α 1 + 2 α + A 0 γ B 1 + B 2 - B 1 3 1 + 2 α , (18) a 3 4 α 2 + 5 α + 2 1 + α 2 1 + 2 α 2 A 0 γ 2 B 1 2 + 1 3 + 4 A 0 γ B 1 + 3 A 0 γ B 1 - B 2 + A 1 γ B 1 3 1 + 2 α + 2 A 0 γ B 1 1 + 2 α 3 α 2 + 3 α + 1 B 1 2 γ 2 1 + 2 α 2 1 + α 2 + A 0 γ B 1 + B 1 - B 2 1 + 2 α .

Proof.

As f C Σ ( α , γ , ϕ ) , so by Definition 1 and using the concept of quasi-subordination, there exist Schwarz functions r ( z ) and s ( z ) and analytic function k ( z ) such that (19) 1 γ 1 - α z f z g z + α z f z g z - 1 = k z ϕ r z - 1 , (20) 1 γ 1 - α w h w j w + α w h w j w - 1 = k w ϕ s w - 1 where r ( z ) = 1 + r 1 z + r 2 z 2 + and s ( w ) = 1 + s 1 w + s 2 w 2 + .

Define the functions p ( z ) and q ( z ) by (21) r z = p z - 1 p z + 1 = 1 2 c 1 z + c 2 - c 1 2 2 z 2 + , (22) s z = q z - 1 q z + 1 = 1 2 d 1 z + d 2 - d 1 2 2 z 2 + . Using (21) and (22) in (19) and (20), respectively, it yields (23) 1 γ 1 - α z f z g z + α z f z g z - 1 = k z ϕ p z - 1 p z + 1 - 1 , (24) 1 γ 1 - α w h w j w + α w h w j w - 1 = k w ϕ q w - 1 q w + 1 - 1 . But (25) 1 γ 1 - α z f z g z + α z f z g z - 1 = 1 γ 1 + α 2 a 2 - b 2 z + 1 + 2 α 3 a 3 - b 3 + 1 + 3 α b 2 2 - 2 a 2 b 2 z 2 + , (26) 1 γ 1 - α w h w j w + α w h w j w - 1 = 1 γ 1 + α b 2 - 2 a 2 w + 1 + 2 α 2 3 a 2 2 - b 2 2 - 3 a 3 - b 3 + 1 + 3 α b 2 2 - 2 a 2 b 2 w 2 + .

Again using (9) and (10) in (21) and (22), respectively, we get (27) k z ϕ p z - 1 p z + 1 - 1 = 1 2 A 0 B 1 c 1 z + 1 2 A 1 B 1 c 1 + 1 2 A 0 B 1 c 2 - c 1 2 2 + A 0 B 2 c 1 2 4 z 2 + , (28) k w ϕ q w - 1 q w + 1 - 1 = 1 2 A 0 B 1 d 1 w + 1 2 A 1 B 1 d 1 + 1 2 A 0 B 1 d 2 - d 1 2 2 + A 0 B 2 d 1 2 4 w 2 +

Using (25) and (27) in (23) and equating the coefficients of z and z 2 , we get (29) 1 + α γ 2 a 2 - b 2 = 1 2 A 0 B 1 c 1 , (30) 1 + 2 α 3 a 3 - b 3 + 1 + 3 α b 2 2 - 2 a 2 b 2 γ = 1 2 A 1 B 1 c 1 + 1 2 A 0 B 1 c 2 - c 1 2 2 + A 0 B 2 4 c 1 2 . Again using (26) and (28) in (24) and equating the coefficients of w and w 2 , we get (31) 1 + α γ b 2 - 2 a 2 = 1 2 A 0 B 1 d 1 , (32) 1 + 2 α 2 3 a 2 2 - b 2 2 - 3 a 3 - b 3 + 1 + 3 α b 2 2 - 2 a 2 b 2 γ = 1 2 A 1 B 1 d 1 + 1 2 A 0 B 1 d 2 - d 1 2 2 + A 0 B 2 4 d 1 2 .

From (29) and (31), it is clear that (33) c 1 = - d 1 , (34) a 2 = A 0 B 1 c 1 γ 4 1 + α + b 2 2 = - A 0 B 1 d 1 γ 4 1 + α + b 2 2 . Therefore on applying triangle inequality and using Lemma 3, (34) yields (35) a 2 1 2 A 0 γ B 1 1 + α + b 2 . As g ( z ) is starlike, so it is well known that b 2 2 , (35) gives (36) a 2 1 2 A 0 γ B 1 1 + α + 2 . Adding (30) and (32), it yields (37) a 2 2 = - 2 α 6 1 + 2 α b 2 2 + 4 1 + 3 α 6 1 + 2 α a 2 b 2 + A 0 B 1 c 2 + d 2 γ 12 1 + 2 α + A 0 B 2 - B 1 c 1 2 + d 1 2 γ 24 1 + 2 α . Using (36) and on applying triangle inequality in (37), we obtain (38) a 2 2 1 + 4 α 3 1 + 2 α b 2 2 + 1 + 3 α A 0 γ B 1 b 2 3 1 + α 1 + 2 α + A 0 γ B 1 + B 2 - B 1 3 1 + 2 α . As g ( z ) is starlike, so using b 2 2 in (38), it yields (39) a 2 4 1 + 4 α 3 1 + 2 α + 2 1 + 3 α A 0 γ B 1 3 1 + α 1 + 2 α + A 0 γ B 1 + B 2 - B 1 3 1 + 2 α . So, result (17) can be easily obtained from (36) and (39).

Now subtracting (32) from (30), we obtain (40) a 3 = b 3 - b 2 2 3 + a 2 2 + A 1 B 1 c 1 - d 1 + A 0 B 1 c 2 - d 2 12 1 + 2 α γ . Applying triangle inequality and using Lemma 3 in (40), it yields (41) a 3 b 3 - b 2 2 3 + a 2 2 + A 1 γ + A 0 γ B 1 3 1 + 2 α . Again adding (30) and (32) and applying triangle inequality, we get (42) a 2 2 2 A 0 γ B 1 1 + 2 α A 0 γ B 1 1 + 2 α + 3 α 2 + 3 α + 1 B 1 2 γ 2 1 + 2 α 2 1 + α 2 + A 0 γ B 1 + B 1 - B 2 1 + 2 α + α 1 + α 2 1 + 2 α A 0 γ 2 B 1 2 + A 0 γ B 1 + B 1 - B 2 1 + 2 α . Using (42) in (41), it gives (43) a 3 4 α 2 + 5 α + 2 1 + α 2 1 + 2 α 2 A 0 γ 2 B 1 2 + b 3 - b 2 2 3 + 4 A 0 γ B 1 + 3 A 0 γ B 1 - B 2 + A 1 γ B 1 3 1 + 2 α + 2 A 0 γ B 1 1 + 2 α 3 α 2 + 3 α + 1 B 1 2 γ 2 1 + 2 α 2 1 + α 2 + A 0 γ B 1 + B 1 - B 2 1 + 2 α . On applying Lemma 4 in (43), the result (18) is obvious.

For α = 0 , Theorem 6 gives the following result.

Corollary 7.

If f ( z ) C Σ ( 0 , γ , ϕ ) , then (44) a 2 m i n . 1 2 A 0 γ B 1 + 2 , 4 3 + 2 A 0 γ B 1 3 + A 0 γ B 1 + B 2 - B 1 3 , (45) a 3 2 A 0 γ 2 B 1 2 + 1 3 + 4 A 0 γ B 1 + 3 A 0 γ B 1 - B 2 + A 1 γ B 1 3 + 2 A 0 γ B 1 B 1 2 γ 2 + A 0 γ B 1 + B 1 - B 2 .

3. Coefficient Bounds for the Function Class <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M151"> <mml:msubsup> <mml:mrow> <mml:mi>C</mml:mi></mml:mrow> <mml:mrow> <mml:mi>Σ</mml:mi></mml:mrow> <mml:mrow> <mml:mn>1</mml:mn></mml:mrow> </mml:msubsup> <mml:mo mathvariant="bold">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo mathvariant="bold">,</mml:mo> <mml:mi>γ</mml:mi> <mml:mo mathvariant="bold">,</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula> Theorem 8.

If f C Σ 1 ( α , γ , ϕ ) , then (46) a 2 m i n . 1 2 A 0 γ B 1 1 + α + 1 , 1 + 4 α 3 1 + 2 α + 1 + 3 α A 0 γ B 1 3 1 + α 1 + 2 α + A 0 γ B 1 + B 2 - B 1 3 1 + 2 α , (47) a 3 4 α 2 + 5 α + 2 1 + α 2 1 + 2 α 2 A 0 γ 2 B 1 2 + 1 9 + 4 A 0 γ B 1 + 3 A 0 γ B 1 - B 2 + A 1 γ B 1 3 1 + 2 α + 2 A 0 γ B 1 1 + 2 α 3 α 2 + 3 α + 1 B 1 2 γ 2 1 + 2 α 2 1 + α 2 + A 0 γ B 1 + B 1 - B 2 1 + 2 α .

Proof.

On applying Lemmas 3 and 5 and following the arguments as in Theorem 6, the proof of this theorem is obvious.

On putting α = 0 , Theorem 8 gives the following result.

Corollary 9.

If f ( z ) C Σ 1 ( 0 , γ , ϕ ) , then (48) a 2 m i n . 1 2 A 0 γ B 1 + 1 , 1 3 + A 0 γ B 1 3 + A 0 γ B 1 + B 2 - B 1 3 , (49) a 3 2 A 0 γ 2 B 1 2 + 1 9 + 4 A 0 γ B 1 + 3 A 0 γ B 1 - B 2 + A 1 γ B 1 3 + 2 A 0 γ B 1 B 1 2 γ 2 + A 0 γ B 1 + B 1 - B 2 .

On putting α = 1 , Theorem 8 gives the following result.

Corollary 10.

If f ( z ) C Σ 1 1 , γ , ϕ , then (50) a 2 m i n . 1 2 A 0 γ B 1 2 + 1 , 5 9 + 2 A 0 γ B 1 9 + A 0 γ B 1 + B 2 - B 1 9 , (51) a 3 11 36 A 0 γ 2 B 1 2 + 1 9 + 4 A 0 γ B 1 + 3 A 0 γ B 1 - B 2 + A 1 γ B 1 9 + 2 A 0 γ B 1 3 7 36 B 1 2 γ 2 + A 0 γ B 1 + B 1 - B 2 3 .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Duren P. L. Univalent Functions 1983 New York, NY, USA Springer MR708494 Robertson M. S. Quasi-subordination and coefficient conjecture Bulletin of the American Mathematical Society 1970 76 1 9 10.1090/S0002-9904-1970-12356-4 MR0251210 Altintas O. Owa S. Mazorizations and quasi-subordinations for certain analytic functions Proceedings of the Japan Academy, Series A, Mathematical Sciences 1992 68 7 181 185 10.3792/pjaa.68.181 MR1193177 Lee S. Y. Quasi-subordinate functions and coefficient conjectures Journal of the Korean Mathematical Society 1975 12 1 43 50 MR0387572 Magesh N. Balaji V. K. Yamini J. Certain subclasses of bistarlike and biconvex functions based on quasi-subordination Abstract and Applied Analysis 2016 2016 6 3102960 10.1155/2016/3102960 MR3498047 Ren F. Y. Owa S. Fukui S. Some inequalities on quasi-subordinate functions Bulletin of the Australian Mathematical Society 1991 43 2 317 324 10.1017/S0004972700029117 MR1097074 Zbl0712.30021 2-s2.0-84971790161 Lewin M. On a coefficient problem for bi-univalent functions Proceedings of the American Mathematical Society 1967 18 63 68 MR0206255 10.1090/S0002-9939-1967-0206255-1 Zbl0158.07802 Brannan D. A. Taha T. S. Mazhar S. M. Hamoni A. Faour N. S. On some classes of bi-univalent functions Mathematical Analysis and Its Applications 1985 3 Oxford, UK Pergamon Press 53 60 KFAS Proceedings Series See also Studia Univ. Babes-Bolyai Math., 31 (1986), No. 2, 70-77 Sakar F. M. Guney H. O. Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion Turkish Journal of Mathematics 2017 41 4 888 895 10.3906/mat-1605-117 MR3683476 Seker B. Sumer Seker S. On subclasses of bi-close-to-convex functions related to the odd-starlike functions Palestine Journal of Mathematics 2017 6 215 221 MR3682521 Selvaraj C. Kumar T. R. K. Bi-univalent coefficient estimates for certain subclasses of close-to-convex functions International Journal of Mathematics and Its Applications 2015 3 4-D 69 74 Pommerenke C. Univalent Functions 1975 Göttingen, Germany Vandenhoeck & Ruprecht MR0507768 Keogh F. R. Merkes E. P. A coefficient inequality for certain classes of analytic functions Proceedings of the American Mathematical Society 1969 20 8 12 10.1090/S0002-9939-1969-0232926-9 MR0232926 Zbl0165.09102