Acta Universitatis Apulensis

Lemma 1. [1] [Theorem 4, with p = 1] A sufficient condition for f(z) defined by (1) to be in the class L(A, B, θ; α) is: ∞ X n=2 n(1 + |B|)|an| ≤(B −A)(1 −α) cos θ.

Lemma 2. [1] [Theorem 1, with p = 1] A sufficient condition for f(z) defined by (1) to be in the class L(A, B, θ; α) is: |an| ≤(B −A)(1 −α) cos θ n (n ≥2).

Lemma 3. [7] Let f(z) ∈A. For some k, the following inequality ∞ X n=2 n(n −1)|an| ≤ 1 k + 2, holds, then f ∈UCV(k). The number 1 k+2 cannot be increased.

Lemma 4. [6] Let f(z) ∈A. For some k, the following inequality ∞ X n=2 (n + k(n −1))|an| ≤1, holds, then f ∈T S(k). 92

Theorem 5. The sufficient condition for H(m; z) to be in the class L(A, B, θ; α) is m −e−m + 1 ≤(B −A)(1 −α) cos θ 1 + |B| . (3)

Corollary 6. Let A = −1 and B = 1 in Theorem 5, then the sufficient condition for H(m; z) to be in the class L(θ; α) is m −e−m + 1 ≤(1 −α) cos θ.

Corollary 7. Let α = 0 in Theorem 5, then the sufficient condition for H(m; z) to be in the class L(A, B, θ) is m −e−m + 1 ≤(B −A) cos θ 1 + |B| . 93

Corollary 8. Let A = −β, B = β, θ = 0 and α = 0 in Theorem 5, then the sufficient condition for H(m; z) to be in the class L(−β, β, 0; 0) = D(β) is m −e−m + 1 ≤ 2β 1 + |β|.

Corollary 9. Let A = −β, B = β and θ = 0 in Theorem 5, then the sufficient condition for H(m; z) to be in the class R(β; α) is m −e−m + 1 ≤2β(1 −α) 1 + |β| .

Theorem 10. The sufficient condition for ψ(m, µ; z) to be in the class L(A, B, θ; α) is µm2 + (1 + 2µ)m −e−m + 1 ≤(B −A)(1 −α) cos θ 1 + |B| . (4)

Corollary 11. Let A = −1 and B = 1 in Theorem 10, then the sufficient condition for H(m; z) to be in the class L(θ; α) is µm2 + (1 + 2µ)m −e−m + 1 ≤(1 −α) cos θ.

Corollary 12. Let α = 0 in Theorem 10, then the sufficient condition for H(m; z) to be in the class L(A, B, θ) is µm2 + (1 + 2µ)m −e−m + 1 ≤(B −A) cos θ 1 + |B| .

Corollary 13. Let A = −β, B = β, θ = 0 and α = 0 in Theorem 10, then the sufficient condition for H(m; z) to be in the class L(−β, β, 0; 0) = D(β) is µm2 + (1 + 2µ)m −e−m + 1 ≤ 2β 1 + |β|.

Corollary 14. Let A = −β, B = β and θ = 0 in Theorem 10, then the sufficient condition for H(m; z) to be in the class R(β; α) is µm2 + (1 + 2µ)m −e−m + 1 ≤2β(1 −α) 1 + |β| .

Theorem 15. The sufficient condition for N(m, µ, λ; z) to be in the class L(A, B, θ; α) is µλm3+(µ−λ+5µλ)m2+(2µ−2λ+1)m+4µλ−e−m+1 ≤(B −A)(1 −α) cos θ 1 + |B| . (5)

Corollary 16. Let A = −1 and B = 1 in Theorem 15, then the sufficient condition for H(m; z) to be in the class L(θ; α) is µλm3 + (µ −λ + 5µλ)m2 + (2µ −2λ + 1)m + 4µλ −e−m + 1 ≤(1 −α) cos θ.

Corollary 17. Let α = 0 in Theorem 15, then the sufficient condition for H(m; z) to be in the class L(A, B, θ) is µλm3 + (µ −λ + 5µλ)m2 + (2µ −2λ + 1)m + 4µλ −e−m + 1 ≤(B −A) cos θ 1 + |B| .

Corollary 18. Let A = −β, B = β, θ = 0 and α = 0 in Theorem 15, then the sufficient condition for H(m; z) to be in the class L(−β, β, 0; 0) = D(β) is µλm3 + (µ −λ + 5µλ)m2 + (2µ −2λ + 1)m + 4µλ −e−m + 1 ≤ 2β 1 + |β|.

Corollary 19. Let A = −β, B = β and θ = 0 in Theorem 15, then the sufficient condition for H(m; z) to be in the class R(β; α) is µλm3 + (µ −λ + 5µλ)m2 + (2µ −2λ + 1)m + 4µλ −e−m + 1 ≤2β(1 −α) 1 + |β| . 96

Theorem 20. If condition m2 + 3m −e−m + 1 ≤(B −A)(1 −α) cos θ 1 + |B| . (6) holds, then [Xm(f)](z) maps the class S or (S∗) to the class L(A, B, θ; α).

Corollary 21. Let A = −1 and B = 1 in Theorem 20, then [Xm(f)](z) maps the class S or (S∗) to the class L(θ; α) if m2 + 3m −e−m + 1 ≤(1 −α) cos θ is true. 97

Corollary 22. Let α = 0 in Theorem 20, then [Xm(f)](z) maps the class S or (S∗) to the class L(A, B, θ) if m2 + 3m −e−m + 1 ≤(B −A) cos θ 1 + |B| , is true

Corollary 23. Let A = −β, B = β, θ = 0 and α = 0 in Theorem 20, then [Xm(f)](z) maps the class S or (S∗) to the class L(−β, β, 0; 0) = D(β) if m2 + 3m −e−m + 1 ≤ 2β 1 + |β|, is true.

Corollary 24. Let A = −β, B = β and θ = 0 in Theorem 20, then [Xm(f)](z) maps the class S or (S∗) to the class R(β; α) if m2 + 3m −e−m + 1 ≤2β(1 −α) 1 + |β| , is true.

Theorem 25. If condition in Eq. (3) satisfied, then [Xm(f)](z) maps the class K to the class L(A, B, θ; α).

Theorem 26. If condition (B −A)(1 −α)m cos θ ≤ 1 k + 2, (7) holds, then [Xm(f)](z) maps the class L(A, B, θ; α) to the class k −UCV . 98

Corollary 27. Let A = −1 and B = 1 in Theorem 26, then [Xm(f)](z) maps the class L(θ; α) to the class k −UCV if 2(1 −α)m cos θ ≤ 1 k + 2, is true.

Corollary 28. Let α = 0 in Theorem 26, then [Xm(f)](z) maps the class L(A, B, θ) to the class k −UCV if (B −A)m cos θ ≤ 1 k + 2, is true

Corollary 29. Let A = −β, B = β, θ = 0 and α = 0 in Theorem 26, then [Xm(f)](z) maps the class L(−β, β, 0; 0) = D(β) to the class k −UCV if 2βm ≤ 1 k + 2, is true. 99

Corollary 30. Let A = −β, B = β and θ = 0 in Theorem 26, then [Xm(f)](z) maps the class R(β; α) to the class k −UCV if 2β(1 −α)m ≤ 1 k + 2, is true.

Theorem 31. If condition (B −A)(1 −α) cos θ  (k + 1 −k m)(1 −e−m) + ke−m  ≤1, (8) holds, then [Xm(f)](z) maps the class L(A, B, θ; α) to the class k −ST.

Corollary 32. Let A = −1 and B = 1 in Theorem 31, then [Xm(f)](z) maps the class L(θ; α) to the class k −ST if (1 −α) cos θ  (k + 1 −k m)(1 −e−m) + ke−m  ≤1 2, is true.

Corollary 33. Let α = 0 in Theorem 31, then [Xm(f)](z) maps the class L(A, B, θ) to the class k −ST if (B −A) cos θ  (k + 1 −k m)(1 −e−m) + ke−m  ≤1, is true

Corollary 34. Let A = −β, B = β, θ = 0 and α = 0 in Theorem 31, then [Xm(f)](z) maps the class L(−β, β, 0; 0) = D(β) to the class k −ST if  (k + 1 −k m)(1 −e−m) + ke−m  ≤1 2β , is true.

Corollary 35. Let A = −β, B = β and θ = 0 in Theorem 31, then [Xm(f)](z) maps the class R(β; α) to the class k −ST if (1 −α)  (k + 1 −k m)(1 −e−m) + ke−m  ≤1 2β , is true. References [1] M. K. Aouf, On certain subclass of analytic p-valent functions of order alpha, Rend. Mat., 7 (8) (1988), 89-104. [2] T. R. Caplinger and W. M. Causey, A class of univalent functions, Proc. Am. Math. Soc., 39 (1973), 357-361.