Initial successive coefficients for certain classes of univalent functions involving the exponential function

Lemma 1. (See [14]) Let ω(z) = P∞ n=1 ckzk be a Schwarz function. Then, for any real number µ and ν the following sharp estimate holds Ψ(ω) = c3 + µc1c2 + νc3 1 ≤Φ(µ, ν), (2.1)

Lemma 2. (See [8, 9]) Let −2 ≤p1 ≤2 and p2, p3 ∈C. Then there exists a function p ∈P with p(z) = 1 + p1z + p2z2 + p3z3 + · · · (2.3) if and only if 2p2 = p2 1 + (4 −p2 1)x (2.4) and 4p3 = p3 1 + 2(4 −p2 1)p1x −(4 −p2 1)p1x2 + 2(4 −p2 1)(1 −|x|2)y

Lemma 3. (See [7]) For given real numbers a, b, c, let Y (a, b, c) = max z∈D ( a + bz + cz2 + 1 −|z|2). (2.6) If a ≥0 and c ≥0, then Y (a, b, c) =    a + |b| + c |b| ≥2(1 −c), 1 + a + b2

Lemma 4. (See [16]) If µ(z) = 1 + P∞ k=1 µkzk is subordinate to ν = 1 + P∞ k=1 νkzk in D, where ν(z) is univalent in D and ν(D) is convex, then |µn| ≤|ν1| (n ≥1). (2.8)

Lemma 5. Suppose that the sequence {Am}∞ m=2 is defined by      Am = λ (m = 2) , Am = λ m −1

Theorem 1. Suppose that f(z) = z + P∞ n=2 anzn ∈S∗ λe. Then |a2| ≤λ, (3.1) |a3| ≤    1 2λ 0 < λ ≤2 3, 3 4λ2

Theorem 2. If f(z) = z + P∞ n=2 anzn ∈Kλe, then |a2| ≤1 2λ, (3.18) |a3| ≤    1 6λ 0 < λ ≤2 3, 1 4λ2

Theorem 3. Let 0 ≤ˆp ≤2λ and p = ˆp/λ. Suppose that f(z) = z + P∞ n=2 anzn ∈ S∗ λe(ˆp). Then the following sharp inequalities |a3 −a2| ≤    1 16λ  8 + 8p −(3λ + 2)p2 0 ≤p ≤ 8 3λ,

Theorem 3. □ Taking λ = 1 in Theorem 3, we obtain the following result.

Corollary 1. Let 0 ≤p ≤2 and f(z) = z + P∞ n=2 anzn ∈S∗ e(p). Then |a3 −a2| ≤1 16(−5p2 + 8p + 8), (3.44) and |a4 −a3| ≤    1 1152(7p3 + 90p2 −252p + 600) 0 ≤p ≤14 9 ,

Theorem 4. Let 0 ≤ˆp ≤λ and p = ˆp/λ. Suppose that f(z) = z + P∞ n=2 anzn be in the class Kλe(ˆp). Then the following sharp inequalities |a3 −a2| ≤1 12λ  2 + 6p −(3λ + 2)p2 , (3.49) and |a4 −a3| ≤    Θ1(λ, p)

Corollary 2. Let 0 ≤p ≤1. Suppose that f(z) = z +P∞ n=2 anzn be in the class Ke(p). Then the following sharp inequalities |a3 −a2| ≤1 12(−5p2 + 6p + 2), (3.67) and |a4 −a3| ≤    1 576 7p3 + 51p2 −72p + 96

Theorem 5. If f(z) = z + P∞ n=2 anzn ∈S∗ λe, then |an| ≤ 1 (n −1)! n−2 Y k=0 (λ + k) (n ≥2). (3.72)

Theorem 6. If f(z) = z + P∞ n=2 anzn ∈Kλe, then |an| ≤1 n! n−2 Y k=0 (λ + k) (n ≥2). Acknowledgments L. Shi was supported by the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2019GGJS195 and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant no. 20B110001 of the P. R. China. Z.-G. Wang was supported by the Key Project of Education Department of Hunan Province under Grant no. 19A097 of the P