JAMSI, 18 (2022), No. 2

LEMMA 1. [24; 11] If p ∈P, then |pk| ≤2,k ∈N, p2 − p2 1 2 ≤2−|p1|2 2 , |pi+j −µ pip j| ≤2,0 ≤µ ≤1, |pn+2k −λ pnp2 k| ≤2(1+2λ),(λ ∈R), |pmpn −pkpl| ≤4,(m+n = k +l;m,n ∈N), and for complex number ρ, we have |p2 −ρ p2 1| ≤2max{1,|2ρ −1|}.

LEMMA 2. Let p ∈P, then |Jp3 1 −Kp1p2 +Lp3| ≤2|J|+2|K −2J|+2|J −K +L|. In particular, it is proved in [24] that |p3 1 −2p1p2 + p3| ≤2.

LEMMA 3. [14; 15] If p ∈P, then 2p2 = p2 1 +(4−p2 1)x, 4p3 = p3 1 +2p1(4−p2 1)x−p1(4−p2 1)x2 +2(4−p2 1)(1−|x|2)z, for |x| ≤1 and |z| ≤1. 2. COEFFICIENT INEQUALITIES

THEOREM 1. If f ∈Rα(ez), then |a2| ≤ 1 1+α , (2) |a3| ≤ 1 1+2α , (3) |a4| ≤ 1 1+3α , (4) and |a5| ≤

COROLLARY 1. If f ∈R ′(ez), then |a2| ≤1,|a3| ≤1,|a4| ≤1,|a5| ≤37 24. For α = 1, Theorem 1 gives the following result due to Kumar and Sharma [12]:

COROLLARY 2. If f ∈R(ez), then |a2| ≤1 2,|a3| ≤1 3,|a4| ≤1 4,|a5| ≤37 120.

THEOREM 2. If f ∈Rα(ez), then |a3 −a2 2| ≤ 1 1+2α . (17) PROOF. From (10) and (11), we have |a3 −a2 2| = 1 2(1+2α) p2 −α2 +6α +3 4(1+α)2 p2 1 .

COROLLARY 3. If f ∈R ′(ez), then |a3 −a2 2| ≤1. Putting α = 1, Theorem 2 yields the following result due to Kumar and Sharma [12]:

COROLLARY 4. If f ∈R(ez), then

THEOREM 3. If f ∈Rα(ez), then |a2a3 −a4| ≤ 1 1+3α . (20) PROOF. Using (10), (11), (12) and after simplification, we have |a2a3 −a4| = 1 48(1+α)(1+2α)(1+3α) (4+12α +2α2)p3 1− (24+72α +24α2)p1p2 +24(1+α)(1+2α)p3 . (21) On applying Lemma 2 in (21), it yields (20).

COROLLARY 5. If f ∈R ′(ez), then |a2a3 −a4| ≤1. On putting α = 1 in Theorem 3, we can obtain the following result due to Kumar and Sharma [12]:

COROLLARY 6. If f ∈R(ez), then |a2a3 −a4| ≤1 4.

THEOREM 4. If f ∈Rα(ez), then |a2a4 −a2 3| ≤ 1 (1+2α)2 . (22) The bound is sharp. PROOF. Using (10), (11) and (12), we have |a2a4 −a2 3| = 1 192(1+α)(1+2α)2(1+3α) 48(1+2α)2p1p3 −24α2p2 1p2+ (−1−4α −α2)p4

COROLLARY 7. If f ∈R ′(ez), then |a2a4 −a2 3| ≤1. Substituting for α = 1 in Theorem 4, the following result due to Kumar and Sharma [12], is obvious:

COROLLARY 8. If f ∈R(ez), then |a2a4 −a2 3| ≤1 9.

THEOREM 5. If f ∈Rα(ez), then |H3(1)| ≤85+850α +3025α2 +4428α3 +2100α4) 24(1+2α)3(1+3α)2(1+4α) . (23)

COROLLARY 9. If f ∈R ′(ez), then |H3(1)| ≤85 24. For α = 1, Theorem 5 yields the following result:

COROLLARY 10. If f ∈R(ez), then |H3(1)| ≤437 2160. 3. BOUNDS OF |H3(1)| FOR TWO-FOLD AND THREE-FOLD SYMMETRIC FUNCTIONS A function f is said to be n-fold symmetric if is satisfy the following condition: f(ξz) = ξ f(z) where ξ = e 2πi n and z ∈E. By S(n), we denote the set of all n-fold symmetric functions which belong to the class S. The n-fold univalent function have the following Taylor-Maclaurin series: f(z) = z+

THEOREM 6. If f ∈Rα(2)(ez), then |H3(1)| ≤ 1 (1+2α)(1+4α). (26)

COROLLARY 11. If f ∈R ′(2)(ez), then |H3(1)| ≤1. For α = 1, Theorem 6 agrees with the following result:

COROLLARY 12. If f ∈R(2)(ez), then |H3(1)| ≤1 15.

THEOREM 7. If f ∈Rα(3)(ez), then |H3(1)| ≤ 1 (1+3α)2 . (32) PROOF. If f ∈Rα(3)(ez), so there exists a function p ∈P(3) such that (1−α) f(z) z +α f ′(z) = e  p(z)−1 p(z)+1  . (33)

COROLLARY 13. If f ∈R ′(3)(ez), then |H3(1)| ≤1. For α = 1, Theorem 7 yields the following result:

COROLLARY 14. If f ∈R(3)(ez), then |H3(1)| ≤1 16. CONCLUSION This paper is concerned with the upper bound of third Hankel determinant for a subclass of analytic functions, associated with exponential function. The results established here are very interesting and will motivate other researchers in this field to work on the similar classes by associating to other standard functions.