Certain Subclasses of Bi-Close-to-Convex Functions Associated with Quasi-Subordination

and Applied Analysis 3 Lemma 5 (see [13]). If g(z) = z + ∑∞k=2 bkzk is a convex function, then 󵄨󵄨󵄨󵄨󵄨b3 − b2 2 󵄨󵄨󵄨󵄨󵄨 ≤ 1 3 . (16) Along with the above lemmas, the following well known results are very useful to derive our main results. Let g(z) = z+∑∞k=2 bkzk be an analytic function inA of the form (1), then |bn| ≤ n, if g(z) is starlike and |bn| ≤ 1, if g(z) is convex. 2. Coefficient Bounds for the Function Class CΣ(α,γ,φ) Theorem 6. If f ∈ CΣ(α, γ, φ), then 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ min. [[ 1 2 [ 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 1 + α + 2] , √ 4 (1 + 4α) 3 (1 + 2α) + 2 (1 + 3α) 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 3 (1 + α) (1 + 2α) + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 3 (1 + 2α) ]] , (17) 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ (4α2 + 5α + 2) (1 + α)2 (1 + 2α)2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 13 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 3 (1 + 2α) + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 (1 + 2α) ⋅ √ (3α2 + 3α + 1)B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 (1 + 2α)2 (1 + α)2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨) (1 + 2α) . (18) Proof. As f ∈ CΣ(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 1 γ [[(1 − α) zf󸀠 (z) g (z) + α(zf 󸀠 (z))󸀠 g󸀠 (z) − 1]] = k (z) (φ (r (z) − 1) , (19) 1 γ [[(1 − α) wh󸀠 (w) j (w) + α(wh 󸀠 (w))󸀠 j󸀠 (w) − 1]] = k (w) (φ (s (w)) − 1) (20) where r(z) = 1+r1z+r2z2+. . . and s(w) = 1+s1w+s2w2+. . .. Define the functions p(z) and q(z) by r (z) = p (z) − 1 p (z) + 1 = 12 [c1z + (c2 − c 2 1 2 ) z2 + . . .] , (21) s (z) = q (z) − 1 q (z) + 1 = 12 [d1z + (d2 − d 2 1 2 ) z2 + . . .] . (22) Using (21) and (22) in (19) and (20), respectively, it yields 1 γ [[(1 − α) zf󸀠 (z) g (z) + α(zf 󸀠 (z))󸀠 g󸀠 (z) − 1]] = k (z) [φ(p (z) − 1 p (z) + 1) − 1] , (23) 1γ [[(1 − α) wh󸀠 (w) j (w) + α(wh 󸀠 (w))󸀠 j󸀠 (w) − 1]] = k (w) [φ(q (w) − 1 q (w) + 1) − 1] . (24) But 1 γ [[(1 − α) zf󸀠 (z) g (z) + α(zf 󸀠 (z))󸀠 g󸀠 (z) − 1]] = 1 γ [(1 + α) ⋅ (2a2 − b2) z + ((1 + 2α) (3a3 − b3) + (1 + 3α) (b2 2 − 2a2b2)) z2 + . . .] , (25) 1 γ [[(1 − α) wh󸀠 (w) j (w) + α(wh 󸀠 (w))󸀠 j󸀠 (w) − 1]] = 1 γ [(1 + α) (b2 − 2a2) w + ((1 + 2α) [2 (3a2 2 − b2 2 ) − (3a3 − b3)] + (1 + 3α) (b2 2 − 2a2b2))w2 + . . .] . (26) Again using (9) and (10) in (21) and (22), respectively, we get k (z) [φ(p (z) − 1 p (z) + 1) − 1] = 12A0B1c1z + [1 2A1B1c1 + 12A0B1 (c2 − c 2 1 2 ) + A0B2c 2 1 4 ] z2 + . . . , (27) k (w) [φ(q (w) − 1 q (w) + 1) − 1] = 12A0B1d1w + [1 2A1B1d1 + 12A0B1 (d2 − d 2 1 2 ) + A0B2d 2 1 4 ] ⋅ w2 + . . . (28) Using (25) and (27) in (23) and equating the coefficients of z and z2, we get (1 + α) γ (2a2 − b2) = 12A0B1c1, (29) 4 Abstract and Applied Analysis (1 + 2α) (3a3 − b3) + (1 + 3α) (b2 2 − 2a2b2) γ = 1 2A1B1c1 + 12A0B1(c2 − c 2 1 2 ) + A0B2 4 c2 1 . (30) Again using (26) and (28) in (24) and equating the coefficients of w and w2, we get (1 + α) γ (b2 − 2a2) = 12A0B1d1, (31) (1 + 2α) [2 (3a2 2 − b2 2) − (3a3 − b3)] + (1 + 3α) (b2 2 − 2a2b2) γ = 1 2A1B1d1 + 12A0B1 (d2 − d 2 1 2 ) + A0B2 4 d21. (32) From (29) and (31), it is clear that c1 = −d1, (33) a2 = A0B1c1γ 4 (1 + α) + b2 2 = −A0B1d1γ 4 (1 + α) + b2 2 . (34) Therefore on applying triangle inequality and using Lemma 3, (34) yields 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ 1 2 [ 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 1 + α + 󵄨󵄨󵄨󵄨b2󵄨󵄨󵄨󵄨] . (35) As g(z) is starlike, so it is well known that |b2| ≤ 2, (35) gives 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ 1 2 [ 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 1 + α + 2] . (36) Adding (30) and (32), it yields a2 2 = − 2α 6 (1 + 2α)b2 2 + 4 (1 + 3α) 6 (1 + 2α)a2b2 + A0B1 (c2 + d2) γ 12 (1 + 2α) + A0 (B2 − B1) (c 2 1 + d21) γ 24 (1 + 2α) . (37) Using (36) and on applying triangle inequality in (37), we obtain 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨2 ≤ (1 + 4α) 3 (1 + 2α) 󵄨󵄨󵄨󵄨b2󵄨󵄨󵄨󵄨2 + (1 + 3α) 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 󵄨󵄨󵄨󵄨b2󵄨󵄨󵄨󵄨 3 (1 + α) (1 + 2α) + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 3 (1 + 2α) . (38) As g(z) is starlike, so using |b2| ≤ 2 in (38), it yields 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ √ 4 (1 + 4α) 3 (1 + 2α) + 2 (1 + 3α) 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 3 (1 + α) (1 + 2α) + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 3 (1 + 2α) . (39) So, result (17) can be easily obtained from (36) and (39). Now subtracting (32) from (30), we obtain a3 = b3 − b2 2 3 + a2 2 + A1B1 (c1 − d1) + A0B1 (c2 − d2) 12 (1 + 2α) γ. (40) Applying triangle inequality and using Lemma 3 in (40), it yields 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨󵄨󵄨b3 − b2 2 󵄨󵄨󵄨󵄨󵄨 3 + 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨2 + ( 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨) B1 3 (1 + 2α) . (41) Again adding (30) and (32) and applying triangle inequality, we get 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨2 ≤ 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 (1 + 2α) [[ 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 (1 + 2α) + √ (3α2 + 3α + 1) B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 (1 + 2α)2 (1 + α)2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨) (1 + 2α) ]] + α (1 + α)2 (1 + 2α) 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨) (1 + 2α) . (42) Using (42) in (41), it gives 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ (4α2 + 5α + 2) (1 + α)2 (1 + 2α)2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 󵄨󵄨󵄨󵄨󵄨b3 − b2 2 󵄨󵄨󵄨󵄨󵄨 3 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 3 (1 + 2α) + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 (1 + 2α) ⋅ √ (3α2 + 3α + 1)B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 (1 + 2α)2 (1 + α)2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨) (1 + 2α) . (43) 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) ∈ CΣ(0, γ, φ), then 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ min. [[ 1 2 [󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + 2] , √ 43 + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 3 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 3 ]] , (44) Abstract and Applied Analysis 5 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 13 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 3 + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1√B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨). (45)and Applied Analysis 5 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 13 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 3 + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1√B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨). (45) 3. Coefficient Bounds for the Function Class C1Σ(α,γ,φ) Theorem 8. If f ∈ C1Σ(α, γ, φ), then 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ min. [[ 12 [ 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 1 + α + 1] , √ (1 + 4α) 3 (1 + 2α) + (1 + 3α) 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 3 (1 + α) (1 + 2α) + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 3 (1 + 2α) ]] , (46) 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ (4α2 + 5α + 2) (1 + α)2 (1 + 2α)2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 19 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 3 (1 + 2α) + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 (1 + 2α) ⋅ √ (3α2 + 3α + 1) B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 (1 + 2α)2 (1 + α)2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨) (1 + 2α) . (47) 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) ∈ C1Σ(0, γ, φ), then 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ min. [[ 12 [󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + 1] , √ 13 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 3 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 3 ]] , (48) 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 19 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 3 + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1√B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨). (49) On putting α = 1, eorem 8 gives the following result. Corollary 10. If f(z) ∈ C1Σ(1, γ, φ), then 󵄨󵄨󵄨󵄨a2󵄨󵄨󵄨󵄨 ≤ min. [[ 1 2 [ 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 2 + 1] , √ 59 + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 9 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B2 − B1󵄨󵄨󵄨󵄨) 9 ]] , (50) 󵄨󵄨󵄨󵄨a3󵄨󵄨󵄨󵄨 ≤ 11 36 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨2 B21 + 19 + 4 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 + (3 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨A1γ󵄨󵄨󵄨󵄨 B1) 9 + 2 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 B1 3 ⋅ √ 7 36B21 󵄨󵄨󵄨󵄨γ󵄨󵄨󵄨󵄨2 + 󵄨󵄨󵄨󵄨A0γ󵄨󵄨󵄨󵄨 (B1 + 󵄨󵄨󵄨󵄨B1 − B2󵄨󵄨󵄨󵄨) 3 . (51) 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.


Introduction and Preliminaries
Let  be the class of functions of the form which are analytic in the open unit disc  = { : || < 1}.
Further, let  be the class of functions () ∈  and univalent in .
By , we denote the class of bounded or Schwarz functions () satisfying (0) = 0 and |()| ≤ 1 which are analytic in the unit disc  and of the form Firstly, it is necessary to recall some fundamental definitions to acquaint with the main content: , the class of quasi-convex functions. ( The functions in the class  are invertible but their inverse function may not be defined on the entire unit disc .The Koebe-one-quarter theorem [1] ensures that the image of  under every function  ∈  contains a disc of radius 1/4.Thus every univalent function  has an inverse  −1 , defined by where A function  ∈  is said to be biunivalent in  if both  and  −1 are univalent in U. Accordingly, a function  ∈  is said to be bistarlike, biconvex, bi-close-to-convex, or bi-quasi-convex if both  and  −1 are starlike, convex, close-to-convex, or quasi-convex respectively. Let Σ denote the class of biunivalent functions in  given by (1).Examples of functions in the class and so on.However, the familiar Koebe function () = /(1 − ) 2 is not a member of Σ.
Let  and  be two analytic functions in .Then  is said to be subordinate to  (symbolically  ≺ ) if there exists a bounded function () ∈  such that () = (()).This result is known as principle of subordination.
Robertson [2] introduced the concept of quasisubordination in 1970.For two analytic functions  and , the function  is said to be quasi-subordinate to  (symbolically  ≺  ) if there exist analytic functions  and  with |()| ≤ 1, (0) = 0 and |()| < 1 such that or equivalently Particularly if () = 1, then () = (()), so that () ≺ () in .So it is obvious that the quasi-subordination is a generalization of the usual subordination.The work on quasisubordination is quite extensive which includes some recent investigations [3][4][5][6].Lewin [7] investigated the class Σ of biunivalent functions and obtained the bound for the second coefficient.Brannan and Taha [8] 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 [9][10][11].
Motivated by earlier work on bi-close-to-convex and quasi-subordination, we define the following subclasses.
For  = 0, the class  Σ (, , ) reduces to  Σ (, ), the class of bi-close-to-convex functions of complex order  defined by quasi-subordination.
It is interesting to note that, for  = 0,  1 Σ (0, , ) is the subclass of bi-close-to-convex functions of complex order  defined by quasi-subordination.Also for  = 1,  1 Σ (1, , ) is the class of bi-quasi-convex functions of complex order  defined by quasi-subordination.
For deriving our main results, we need the following lemmas.
Along with the above lemmas, the following well known results are very useful to derive our main results.Let () = +∑ ∞ =2     be an analytic function in  of the form ( ), then