Convexity properties for some general integral operators

Abstract. For analitic functions f and g in the open unit disk U. In this paper we consider the classes N(β), K(β) and two integral operarors functions fn(f, g) and Jn(f, g), where g is a functions that the belongs to the family B(µ, β). The main object of this paper is to obtain some properties in this class and and the order of convexity for two integral operators. Keywords: Analytic functions, univalent functions, starlike functions, convex functions, integral operators.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0026 Natural Science, 2018, Volume 63, Issue 6, pp. 30-38 This paper is available online at CONVEXITY PROPERTIES FOR SOME GENERAL INTEGRAL OPERATORS Nguyen Van Tuan1 and Daniel Breaz2 1Department of Mathematics, University of Pitesti, Romania 2"1 Decembrie" University of Alba Iulia, Romania Abstract. For analitic functions f and g in the open unit disk U . In this paper we consider the classes N (β), K(β) and two integral operarors functions fn(f, g) and Jn(f, g), where g is a functions that the belongs to the family B(µ, β). The main object of this paper is to obtain some properties in this class and and the order of convexity for two integral operators. Keywords: Analytic functions, univalent functions, starlike functions, convex functions, integral operators. 1. Introduction and preliminaries Let U = {z : |z| < 1} be the open unit disk and A be the class of all functions of the form f(z) = z + ∞∑ n=2 anz n, z ∈ U , (1.1) which are analytic in U and satisfy the condition f(0) = f ′ (0)− 1 = 0. We denote by S the class of univalent and regular functions. A function f ∈ A is a starlike function of order β, 0 ≤ β < 1 and we denote this class by S∗(β) if it satisfies (see in [1]) Re ( zf ′ (z) f(z) ) > β, z ∈ U . (1.2) We denote by K(β) the class of convex functions of order β, 0 ≤ β < 1 that satisfies the inequality (see in [2]) Re ( zf ′′ (z) f ′(z) + 1 ) > β, ∣∣∣∣∣zf ′′ (z) f ′(z) ∣∣∣∣∣ < 1− β, z ∈ U . (1.3) A function f ∈ A belongs to class R(β), 0 ≤ β < 1 if and only if Re(f ′ (z)) > β, z ∈ U . (1.4) Received August 10, 2018. Revised August 25, 2018. Accepted August 30, 2018. Contact Nguyen Van Tuan, e-mail: vataninguyenedu@gmail.com 30 Convexity properties for some general integral operators LetN (β) be the subclass ofA that contains all the functions f , which satisfy the inequality Re ( zf ′′ (z) f ′(z) + 1 ) 1, z ∈ U . (1.5) A. Uralegaddi, M. D. Ganigi and S. M. Sarangi in [3] and S. Owa and H. M. Srivastava in [4] introduced and studied the class N (β). The family B(µ, β), µ ≥ 0, 0 ≤ β < 1, which contains the function f(z) that satisfy the condition ∣∣∣∣f ′(z) ( z f(z) )µ − 1 ∣∣∣∣ < 1− β, z ∈ U (1.6) was studied by B. A. Frasin and J. Jahangiri in [5]. The family B(µ, β) is a comprehensive class of analytic functions that includes various new classes of analytic univalent functions, such as B(1, β) ≡ S∗(β) and B(0.β) ≡ R(β). The subclass B(2, β) ≡ B(β) has been introduced by B. A. Frasin and M. Darus in [6]. Lemma 1.1. [7] (General Schwarz-Lemma). Let f the function regular in the disk UR = {z ∈ C : |z| < R}, with |f(z)| < M , M fixed. If f has at z = 0 one zero with multiply ≥ m, then |f(z)| ≤ M Rm |z|m, z ∈ UR (1.7) the equality (in the inequality (1.7) for z 6= 0) can hold only if f(z) = eiθ M Rm zm, (1.8) where θ is constant. In this paper, we study general integral operator In(f, g)(z), defined by A. Oprea and D. Breaz in [8] In(f, g)(z) = ∫ z 0 ( f ′ (tn)eg(t) )α dt (1.9) and we define a new general integral operator Jn(f, g)(z) Jn(f, g)(z) = ∫ z 0 ( f ′ (tn) )α ( eg(t) )λ dt (1.10) is in the class N(ρ) and K(ρ) by using functions from the class B(µ, β). 2. Main results Theorem 2.1. Let the functions f,∈ A, g ∈ B(µ, β), µ ≥ 1, 0 ≤ β < 1 and α ∈ C , Reα ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator In(f, g) defined by (1.9) is in the class N (ρ), where ρ = |α|[n + (2− β)Mµ] + 1, n ∈ N∗. (2.1) 31 Nguyen Van Tuan and Daniel Breaz Proof. From (1.9), we have I ′ n(f, g)(z) = ( f ′ (zn)eg(z) )α (2.2) and I ′′ n(f, g)(z) = α ( f ′ (zn)eg(z) )α−1 [ nzn−1f ′′ (zn)eg(z) + f ′ (zn)eg(z)g ′ (z) ] (2.3) From (2.2) and (2.3), we get zI ′′ n(f, g)(z) I ′ n(f, g)(z) = α [ nzn f ′′ (zn) f ′(zn) + zg ′ (z) ] (2.4) so, we have Re ( zI ′′ n(f, g)(z) I ′ n(f, g)(z) + 1 ) = Re ( α [ nzn f ′′ (zn) f ′(zn) + zg ′ (z) ] + 1 ) (2.5) Since Reω ≤ |ω| and from (2.5), we get Re ( zI ′′ n(f, g)(z) I ′ n(f, g)(z) + 1 ) ≤ ∣∣∣∣∣α [ nzn f ′′ (zn) f ′(zn) + zg ′ (z) ] + 1 ∣∣∣∣∣ ≤ |α| [ n|zn| ∣∣∣∣∣f ′′ (zn) f ′(zn) ∣∣∣∣∣+ |z| |g′ (z)| ] + 1 < |α|[n|z|n + |z| |g ′ (z)|] + 1 < |α| [ n|z|n + |z| ∣∣∣∣g′(z) ( z g(z) )µ∣∣∣∣ ∣∣∣∣g(z)z ∣∣∣∣ µ] + 1. (2.6) Since g ∈ B(µ, β), |g(z)| < M , applying the General Schwarz Lemma and from (2.6), we get Re ( zI ′′ n(f, g)(z) I ′ n(f, g)(z) + 1 ) < |α| [ n|z|n + |z| ∣∣∣∣g′(z) ( z g(z) )µ∣∣∣∣ ∣∣∣∣g(z)z ∣∣∣∣ µ] + 1 < |α| [ n|z|n + |z| (∣∣∣∣g′(z) ( z g(z) )µ − 1 ∣∣∣∣+ 1 ) ∣∣∣∣g(z)z ∣∣∣∣ µ] + 1 < |α|[n|z|n + |z|(2 − β)Mµ] + 1. (2.7) Let us define the function G : [0, 1]→ R, G(x) = nxn + x(2− β)Mµ, x = |z|, n ∈ N∗. We see that, G(x) is a continuous function and increase for all x ∈ [0, 1], so, the maximum of function G(x) is G(1). So, from (2.7) we obtain Re ( zI ′′ n(f, g)(z) I ′ n(f, g)(z) + 1 ) < |α|[n + (2− β)Mµ] + 1 = ρ. (2.8) From (2.8), we obtain the integral operator In(f, g) is in the class N (ρ). 32 Convexity properties for some general integral operators Corollary 2.1. Let the functions f ∈ A, with g ∈ S∗(β), 0 ≤ β < 1, α ∈ C, Reα ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator In(f, g) defined by (1.9) is in the class N (ρ), where ρ = |α|[n + (2− β)M ] + 1. (2.9) Proof. In Theorem 2.1, we put µ = 1 If we consider µ = 0 in Theorem 2.1, we obtain Corollary 2.2. Let the functions f ∈ A, with g ∈ R(β) with 0 ≤ β < 1 and α a complex with Reα ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator In(f, g) defined by (1.9) is in the class N (ρ), where ρ = |α|(n + 2− β) + 1. (2.10) Remark 2.1. If we consider n = 1, α = 1, the integral operator In(f, g) in relation (1.9), we obtain respectively the integral operator I1(f, g)(z) = ∫ z 0 ( f ′ (t)eg(t) )α dt and the integral operator I(f, g)(z) = ∫ z 0 f ′ (t)eg(t)dt, introduced and studied in [9]. Theorem 2.2. Let the functions f,∈ A, with g ∈ B(µ, β) with µ ≥ 1, 0 ≤ β < 1 and α ∈ C, Reα ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator In(f, g) defined by (1.9) is in the class K(ρ), where ρ = 1− |α| (n+ (2− β)Mµ) (2.11) and 0 < |α|(n + (2− β)Mµ) ≤ 1. (2.12) Proof. After the same steps as in the proof of Theorem 2.1, we get zI ′′ n(f, g)(z) I ′ n(f, g)(z) = α ( nzn f ′′ (zn) f ′(zn) + zg ′ (z) ) . (2.13) From (2.13), it follows that ∣∣∣∣∣zI ′′ n(f, g)(z) I ′ n(f, g)(z) ∣∣∣∣∣ = ∣∣∣∣∣α ( nzn f ′′ (zn) f ′(zn) + zg ′ (z) )∣∣∣∣∣ ≤ |α| ( |zn|n ∣∣∣∣∣f ′′ (zn) f ′(zn) ∣∣∣∣∣+ |z| |g′ (z)| ) < |α| ( n|z|n + |z| ∣∣∣∣g′(z) ( z g(z) )µ∣∣∣∣ ∣∣∣∣g(z)z ∣∣∣∣ µ) . (2.14) 33 Nguyen Van Tuan and Daniel Breaz Since g ∈ B(µ, β), |g(z)| < M , applying the General Schwarz Lemma and from (2.14), we obtain∣∣∣∣∣zI ′′ n(f, g)(z) I ′ n(f, g)(z) ∣∣∣∣∣ < |α| ( n|z|n + |z| ∣∣∣∣g′(z) ( z g(z) )µ∣∣∣∣ ∣∣∣∣g(z)z ∣∣∣∣ µ) < |α| ( n|z|n + |z| (∣∣∣∣g′(z) ( z g(z) )µ − 1 ∣∣∣∣+ 1 ) ∣∣∣∣g(z)z ∣∣∣∣ µ) < |α| (n|z|n + |z|(2 − β)Mµ) . (2.15) Let us consider the function G : [0, 1]→ R, G(x) = nxn + x(2− β)Mµ, x = |z|, n ∈ N∗. We see that, G(x) is a continuous function and G ′ (x) > 0 for all x ∈ [0, 1], so, the maximum of function G(x) is G(1). So, from (2.15) we obtain∣∣∣∣∣zI ′′ n(f, g)(z) I ′ n(f, g)(z) ∣∣∣∣∣ < |α| (n+ (2− β)Mµ) = 1− ρ. (2.16) Which implies that In(f, g) í in the class K(ρ). Corollary 2.3. Let the functions f, g ∈ A, with g is in the class S∗(β), 0 ≤ β < 1 and α a complex with Reα ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and∣∣∣∣ f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator In(f, g) defined by (1.9) is in the class K(ρ), where ρ = 1− |α| (n+ (2− β)M) (2.17) and 0 < |α|(n + (2− β)M) ≤ 1. (2.18) Proof. In Theorem 2.2, we put µ = 1. If we consider µ = 0 in Theorem 2.2, we obtain Corollary 2.4. Let the functions f ∈ A, g ∈ R(β), 0 ≤ β < 1, α ∈ C, Reα ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator In(f, g) defined by (1.9) is in the class K(ρ), where ρ = 1− |α|(n + 2− β) (2.19) and 0 < |α|(n + 2− β) ≤ 1. (2.20) Remark 2.2. If we consider n = 1, the integral operator In(f, g) in relation (1.9), we obtain the integral operator I1(f, g)(z) = ∫ z 0 ( f ′ (t)eg(t) )α dt, introduced and studied in [9]. 34 Convexity properties for some general integral operators Theorem 2.3. Let the functions f ∈ A, g ∈ B(µ, β), µ ≥ 1, 0 ≤ β < 1, α, λ ∈ C, Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣ f ′′ (z)f ′(z) ∣∣∣∣ < 1, then the integral operator Jn(f, g) defined by (1.10) is in the class N (ρ), where ρ = n|α|+ |λ|(2− β)Mµ + 1. (2.21) Proof. From (1.10), we have J ′ n(f, g)(z) = ( f ′ (zn) )α ( eg(z) )λ (2.22) and J ′′ n (f, g)(z) = ( f ′ (zn) )α−1 ( eg(z) )λ−1 [ αnzn−1f ′′ (zn)eg(z) + λg ′ (z)f ′ (zn)eg(z) ] . (2.23) From (2.22) and (2.23), we get zJ ′′ n (f, g)(z) J ′ n(f, g)(z) = αnzn f ′′ (zn) f ′ (zn) + λzg ′ (z) (2.24) From (2.24), we obtain Re ( zJ ′′ n (f, g)(z) J ′ n(f, g)(z) + 1 ) = Re ( αnzn f ′′ (zn) f ′(zn) + λzg ′ (z) + 1 ) (2.25) Since Reω ≤ |ω| and from (2.25), we have Re ( zJ ′′ n (f, g)(z) J ′ n(f, g)(z) + 1 ) ≤ ∣∣∣∣∣ ( αnzn f ′′ (zn) f ′(zn) + λzg ′ (z) ) + 1 ∣∣∣∣∣ ≤ n|α||z|n ∣∣∣∣∣f ′′ (zn) f ′(zn) ∣∣∣∣∣+ |λ||z||g′ (z)|+ 1 < n|α||z|n + |λ||z||g ′ (z)| + 1 < n|α||z|n + |λ||z| ∣∣∣∣g′(z) ( z g(z) )µ∣∣∣∣ ∣∣∣∣g(z)z ∣∣∣∣ µ + 1. (2.26) Since g ∈ B(µ, β), |g(z)| < M , aplying the General Schwarz Lemma and from (2.26), we obtain Re ( zJ ′′ n (f, g)(z) J ′ n(f, g)(z) + 1 ) < n|z|n + |λ||z| (∣∣∣∣g′(z) ( z g(z) )µ − 1 ∣∣∣∣+ 1 ) ∣∣∣∣g(z)z ∣∣∣∣+ 1 < n|α||z|n + |λ||z|(2 − β)Mµ + 1. (2.27) Let us define the function G : [0, 1]→ R, G(x) = n|α|xn + |λ|(2 − β)Mµx+ 1, (x = |z|, n ∈ N∗). (2.28) 35 Nguyen Van Tuan and Daniel Breaz We see that, G(x) is continuous function and G ′ (x) > 0 for all x ∈ [0, 1], so, the maximum of function G(x) is G(1). From (2.27) and (2.28), we obtain Re ( zJ ′′ n (f, g)(z) J ′ n(f, g)(z) + 1 ) < n|α|+ |λ|(2− β)Mµ + 1 = ρ. (2.29) From (2.29), we obtain the integral operator Jn(f, g) is in the class N (ρ). Corollary 2.5. Let the functions f ∈ A, g ∈ B(µ, β), µ ≥ 1, 0 ≤ β < 1, α, λ ∈ C Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣ f ′′ (z)f ′(z) ∣∣∣∣ < 1, then the integral operator J1(f, g)(z) = ∫ z 0 ( f ′ (t) )α ( eg(t) )λ dt is in the class N (ρ), where ρ = |α| + |λ|(2− β)Mµ + 1. (2.30) Proof. In Theorem 2.3, we put n = 1 Corollary 2.6. Let the functions f,∈ A, g ∈ S∗(β), 0 ≤ β < 1, α, λ ∈ C, Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator Jn(f, g) defined by (1.10) is in the class N (ρ), where ρ = n|α|+ |λ|(2 − β)M + 1. (2.31) Proof. In Theorem 2.3, we put µ = 1 If we consider µ = 0 in Theorem 2.3, we obtain Corollary 2.7. Let the functions f ∈ A, g ∈ R(β), 0 ≤ β < 1, α, λ ∈ C Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, then the integral operator Jn(f, g) defined by (1.10) is in the class N (ρ), where ρ = n|α|+ |λ|(2 − β) + 1. (2.32) Theorem 2.4. Let the functions f ∈ A, g ∈ B(µ, β), µ ≥ 1, 0 ≤ β < 1, α, λ ∈ C Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣ f ′′ (z)f ′(z) ∣∣∣∣ < 1, then the integral operator Jn(f, g) defined by (1.10) is in the class K(ρ), where ρ = 1− n|α| − |λ|(2 − β)Mµ (2.33) and 0 < n|α|+ |λ|(2 − β)Mµ ≤ 1. (2.34) Proof. After the same steps as in the proof of Theorem 2.3, we get 36 Convexity properties for some general integral operators ∣∣∣∣∣zJ ′′ n (f, g)(z) J ′ n(z) ∣∣∣∣∣ = ∣∣∣∣∣nαzn f ′′ (zn) f ′(zn) + λzg ′ (z) ∣∣∣∣∣ ≤ n|α||z|n + |λ||z||g ′ (z)| ≤ n|α||z|n + |λ||z| ∣∣∣∣g′(z) ( z g(z) )µ∣∣∣∣ ∣∣∣∣g(z)z ∣∣∣∣ µ . (2.35) Since g ∈ B(µ, β), |g(z)| < M , applying the General Schwarz Lemma and from (2.35), we obtain∣∣∣∣∣zJ ′′ n (f, g)(z) J ′ n(z) ∣∣∣∣∣ ≤ n|α||z|n + |λ||z| (∣∣∣∣g′(z) ( z g(z) )µ − 1 ∣∣∣∣+ 1 ) ∣∣∣∣g(z)z ∣∣∣∣ µ ≤ n|α||z|n + |λ||z|(2 − β)Mµ. (2.36) Let us define the function G : [0, 1]→ R, G(x) = n|α|xn + |λ|(2 − β)Mµx, (x = |z|, n ∈ N∗). (2.37) We see that, G(x) is a continuous function and G ′ (x) > 0, for all x ∈ [0, 1], so, the maximum of function G(x) is G(1), so, from (2.36) and (2.37), we obtain∣∣∣∣∣zJ ′′ n (f, g)(z) J ′ n(z) ∣∣∣∣∣ ≤ n|α|+ |λ|(2− β)Mµ = 1− ρ. (2.38) Which implies that Jn(f, g)(z) is in the class K(ρ). If we consider n = 1 in Theorem 2.4, we obtain Corollary 2.8. Let the functions f ∈ A, g ∈ B(µ, β), µ ≥ 1, 0 ≤ β < 1, α, λ ∈ C Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, the integral operator J1(f, g)(z) = ∫ z 0 ( f ′ (t) )α ( eg(t) )λ dt is in the class K(ρ), where ρ = 1− |α| − |λ|(2− β)Mµ (2.39) and 0 < |α|+ |λ|(2− β)Mµ ≤ 1. (2.40) Corollary 2.9. Let the functions f ∈ A, g ∈ S∗(β), 0 ≤ β < 1, α, λ ∈ C Reα ≥ 0, Reλ ≥ 0. If |g(z)| < M , for M a positive real number (M ≥ 1), z ∈ U and ∣∣∣∣f ′′ (z)f ′ (z) ∣∣∣∣ < 1, the integral operator Jn(f, g) defined by (1.10) is in the class K(ρ), where ρ = 1− n|α| − |λ|(2 − β)M (2.41) and 0 < n|α|+ |λ|(2 − β)M ≤ 1. (2.42) 37 Nguyen Van Tuan and Daniel Breaz Proof. In Theorem 2.4, we put µ = 1. If we consider µ = 0 in Theorem 2.4, we obtain Corollary 2.10. Let the functions f ∈ A, g ∈ R(β), 0 ≤ β < 1 and α, λ ∈ C, Reα ≥ 0, Reλ ≥ 0. 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