On sharp bounds of certain Close-to-Convex functions

Lemma 1.1. [11, p.41] If p(z) is in P and is given by (1.2), then |cn|≤2 for each n.

Lemma 1.2. [11] Let p ∈P and is given by (1.2). Then c2 −c2 1 2 ≤2 −|c1|2 2 . This inequality is sharp for the functions Pt,ϑ(z) given by Pt,ϑ(z) = t 1 + eiϑz 1 −eiϑz  + (1 −t) 1 + ei2ϑz2 1 −ei2ϑz2 

Lemma 1.3. [10] Let p ∈P and is given by (1.2). Then |c3 −2c1c2 + c3 1|≤2 and |c4 1 −3c2 1c2 + c2 2 + 2c1c3 −c4|≤2. This result is sharp for the function p(z) = (1 + z)/(1 −z).

Lemma 1.4. [3] Let Ω(A, B, C, M) = max |v|≤1(|M|(1 −|v|2) + |A + Bv + Cv2|). (1.3) If AC ≥0, then Ω(A, B, C, M) =    |A|+|B|+|C|, |B|≥2(|M|−|C|), |M|+|A|+ B2 4(|M|−|C|), |B|< 2(|M|−|C|).

Lemma 1.5. [2] If p ∈P is of the form p(z) = 1 + c1z + c2z2 + c3z3 + · · · , then c1 = 2ζ1 c2 = 2ζ2 1 + 2(1 −|ζ1|2)ζ2 c3 = 2ζ3 1 + 4(1 −|ζ1|2)ζ1ζ2 −2(1 −|ζ1|2) ¯ζ1ζ2 2 + 2(1 −|ζ1|2)(1 −|ζ2|2)ζ3 for some ζi ∈D (i = 1, 2, 3). In view of the same, in the following section we derive the general formula for the fourth

Lemma 2.1. If p ∈P is of the form p(z) = 1 + c1z + c2z2 + c3z3 + · · · , then 1 2c4 = ζ4 1 + (1 −|ζ1|2)(3ζ2 1ζ2 −2ζ2 2|ζ1|2+ ¯ζ1 2ζ3 2) + ζ2 2(1 −|ζ1|2)2 + (1 −|ζ1|2)(1 −|ζ2|2)(2ζ1ζ3 −2 ¯ζ1ζ2ζ3 −¯ζ2ζ2 3 + (1 −|ζ3|2)ζ4) (2.1) for some ζi ∈D.

Theorem 3.1. Let f(z) = z +a2z2 +a3z3 +. . . ∈F1. Then (i) |δ2|≤3/2, (ii) |δ3|≤17/6. Further, if a2 ∈R, then (iii) |δ4|≤49/8, (iv) |δ5|≤1729/120. These bounds are sharp.

Theorem 3.2. Let f(z) = z + a2z2 + a3z3 + . . . ∈F2. Then (i) |δ2|≤1. Further if a2 ∈R, then (ii) |δ3|≤1, (iii) |δ4|≤16/3 √ 15, (iv) |δ5|≤2.947584. Except (iv) rest all above bounds are sharp.

Lemma 1.1, (3.12) reduces to |δ2|≤|c1| 2 ≤1. We know that this inequality is sharp, whenever |c1|= 2 which is true for the function P1,ϑ(z) (0 ≤ ϑ < 2π) given in Lemma 1.2. The upper bound of |δ2| is sharp since there exists an extremal function ˆf2 ∈F2, which is the solution of z ˆf ′2(z) = z(1 −z2)−1P1,ϑ(z) (0 ≤ϑ < 2π). (ii) By taking b2 = 0 and b3 = 1 in (3.9), we get δ3 = 1 6(3c2 1 −2 −2c2). We have a2 ∈R, which together with (3.4) yields that c1 ∈R. Then by using Lemma 1.5, we have ζ1 ∈R an

Theorem 3.3. Let f(z) = z +a2z2 +a3z3 +. . . ∈F3. Then (i) |δ2|≤3/2, (ii) |δ3|≤19/6. Further, if a2 ∈R, then (iii) |δ4|≤61/8, (iv) |δ5|≤2371/120. These bounds are sharp.

Lemma 1.1, (3.12) reduces to 2|δ2|≤1 + |c1|≤3. We know that this inequality is sharp whenever |c1|= 2, which is true for the function P1,ϑ(z) (0 ≤ ϑ < 2π), defined in Lemma 1.2. The inequality |δ2|≤3/2 is sharp since there exists an extremal function ˆf3 ∈F3, which is a solution of z ˆf ′3(z) = z(1 −z)−1P1,ϑ(z). (ii) By taking b2 = 1 and b3 = 0 in (3.14), we get 3|δ3|≤2 −p2 2 + −1 2 −  1 + pq + ip p 1 −q2 2 ,

Theorem 3.4. Let f(z) = z + a2z2 + a3z3 + . . . ∈F4. Then (i) |δ2|≤2, (ii) |δ3|≤5. Further, if a2 ∈R, then (iii) |δ4|≤14, (iv) |δ5|≤42. These bounds are sharp.

Lemma 1.1, (3.12) reduces to 2|δ2|≤2 + |c1|≤4. We know that this inequality is sharp whenever |c1|= 2, which is true for the function P1,ϑ(z) (0 ≤ ϑ < 2π), defined in Lemma 1.2. The inequality |δ2|≤2 is sharp since there exists an extremal function ˆf4 ∈F4, which is a solution of z ˆf ′4(z) = z(1 −z)−1P1,ϑ(z). (ii) By taking b2 = 2 and b3 = 3 in (3.14), we get 3|δ3|≤2 −p2 2 + 1 −(2 + pq + ip p 1 −q2)2 , which can be written as 3|δ3|≤2 −p2 2 + p