Some properties of a new integral operator

3. Conclusion Our present results were motivated essentially by several recent works dealing with the interesting problem of finding convexity of order for some integral operators (see, for example, [2, 7, 8, 10]). In our study here, we have successfully determined the convexity properties for the functions Hn(z) given by the general family of integral operators in Eq. (2.1). For gi ∈ Gbi, 0 < bi ≤ 1 (i = 1, . . . , n), we study the convexity order of Hn (Theorem 2.1) and the next result find the convexity order ρ of Hn, when fi ∈ S∗(βi), hi ∈ S∗(δi), 0 ≤ βi, δi < 1, for i = 1, . . . , n (Theorem 2.2 ). These theorems are serval corollaries (Corollary 2.1 and corollary 2.2). The following results the convexity of Hn, when fi ∈ SP(α, β), hi ∈ SP(δ, η), α > 0, δ > 0, β ∈ [0, 1), η ∈ [0, 1) and gi ∈ N (λi), λi > 1 and when fi ∈ Sλ∗i(b), hi ∈ Sδ∗i(b), gi ∈ Cλi(b), 0 ≤ λi, δi < 1 for i = 1, . . . , n, b ∈ C − {0} (Theorem 2.3 and Theorem 2.4). Also for particular cases we get from these theorems Corollary 2.3 and Corollary 2.4. Also, if gi ∈ N (λi), λi > 1, for i = 1, . . . , n, the last result studies convexity conditions so that Hn ∈ N (µ) (Theorem 2.5 and for a special case Corollary 2.5).

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0025 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 3-14 This paper is available online at SOME PROPERTIES OF A NEW INTEGRAL OPERATOR Nguyen Van Tuan1, Adriana Oprea1 and Daniel Breaz2 1Department of Mathematics, University of Pitesti, Romania 21 Decembrie, University of Alba Iulia, Romania Abstract. For certain classes of analytic functions fi, gi and hi, i = 1, . . . , n in the open unit disk U , we define a new general integral operator Hn(z) =∫ z 0 ∏n i=1 ( fi(t) hi(t) )αi (g′i(t)) γi dt and we study convexity properties of this integral operator. Keywords: Analytic functions, integral operators, convexity order, convex funtions, order. 1. Introduction Let U = {z : |z| < 1} be the unit disk and A be the class of all functions in the form f(z) = z + ∞∑ k=2 akz k, z ∈ U (1.1) which are analytic in U and satisfy the conditions f(0) = f ′(0) − 1 = 0. We denote by S the subclass ofA consisting of univalent functions on U . In [1] and [2] they were defined the following classes of functions. A function f ∈ A is a convex function of complex order b, (b ∈ C \ {0}) and type λ (0 ≤ λ < 1), if and only if Re { 1 + 1 b ( zf ′′(z) f ′(z) )} > λ or ∣∣∣∣∣1b zf ′′ (z) f ′(z) ∣∣∣∣∣ < 1− λ, (z ∈ U). (1.2) We denote by C∗λ(b) the class of these 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 Re ( zf ′(z) f(z) ) > β or ∣∣∣∣∣zf ′ (z) f(z) ∣∣∣∣∣ < β, z ∈ U. (1.3) We denote by K(β) the class of convex functions of oder β, 0 ≤ β < 1 that satisfies the inequality Re ( zf ′′ (z) f ′(z) + 1 ) > β, z ∈ U. (1.4) Received January 21, 2015. Accepted October 19, 2015. Contact Nguyen Van Tuan, e-mail address: nvtuan@moet.edu.vn 3 Nguyen Van Tuan, Adriana Oprea and Daniel Breaz A function f ∈ A belongs to class R(β), 0 ≤ β < 1, if Re ( f ′(z) ) > β, z ∈ U. (1.5) A function f ∈ A is a starlike function of the complex order b, b ∈ C \ {0} and type λ, (0 ≤ λ < 1), if and only if Re { 1 + 1 b ( zf ′(z) f(z) − 1 )} > λ or ∣∣∣∣∣1b zf ′ (z) f(z) ∣∣∣∣∣ ≤ 1− λ, z ∈ U. (1.6) We denote by S∗λ(b) the class of the those functions. F. Ronning introduced in [3] the class of univalent functions SP(α, β), α > 0, β ∈ [0, 1). So, we denote by SP(α, β) the class of all functions f ∈ S, which satisfies the inequality∣∣∣∣zf ′(z)f(z) − (α+ β) ∣∣∣∣ ≤ Rezf ′(z)f(z) + α− β, z ∈ U. (1.7) Silverman defined in [4] the class Gb, so a function f ∈ A is in the class Gb, 0 < b ≤ 1 if and only if ∣∣∣∣1 + zf ′′(z)f ′(z) − zf ′(z)f(z) ∣∣∣∣ < b ∣∣∣∣zf ′(z)f(z) ∣∣∣∣ , z ∈ U. (1.8) Uralegaddi in [5], Owa and Srivastava in [6] defined the class N (β). So, a function f ∈ A is in the class N (β) if it verifies the inequality Re ( zf ′′ (z) f ′(z) + 1 ) 1. (1.9) 2. Main results In this paper, we introduce some properties for a new integral operator defined by Eq. (2.1) Hn(z) = ∫ z 0 n∏ i=1 ( fi(t) hi(t) )αi ( g′i(t) )γi dt (2.1) Remark 2.1. If we consider hi(z) = z, and αi = γi, for i = 1, 2, ..., n, in relation Eq. (2.1), we obtain the integral operator Gn(z) = ∫ z 0 n∏ i=1 ( fi(t) t g ′ i(t) )αi dt (2.2) Remark 2.2. If we consider hi(z) = z, for all i = 1, 2, . . . , n, in Eq. (2.1), we get the integral operator Fn(z) = ∫ z 0 n∏ i=1 ( fi(t) t )αi ( g′i(t) )γi dt (2.3) introduced and studied by D. Breaz and L. Stanciu in [2] and studied by L. Stanciu in [7, 8]. 4 Some properties of a new integral operator Remark 2.3. If fi(z) = z, hi(z) = z, αi = γi, for i = 1, 2, ...n from Eq. (2.1), we obtain the integral operator Fα1,α2,...,αn(z) = ∫ z 0 (g1(t)) α1 .(g2(t)) α2 ...(gn(t)) αndt, (2.4) introduced and studied by D. Breaz et al. in [9]. Remark 2.4. For n = 1, f(z) = z, h(z) = z, g1 = g, α1 = γ1 = γ from Eq. (2.1), we obtain the integral operator G(z) = ∫ z 0 (g ′ (t))γdt (2.5) studied in [10] and [11]. Theorem 2.1. Let fi, gi, hi ∈ A, where gi ∈ Gbi , 0 < bi ≤ 1, for all i = 1, 2, ..., n. For anyMi, Ni ≥ 1, which verify∣∣∣∣∣zf ′ i (z) fi(z) ∣∣∣∣∣ ≤Mi, ∣∣∣∣∣zh ′ i(z) hi(z) ∣∣∣∣∣ ≤ Ni and ∣∣∣∣∣zg ′ i(z) gi(z) − 1 ∣∣∣∣∣ < 1, for all z ∈ U , there are real numbers αi > 0, with i = 1, 2, ...n so that λ = 1− n∑ i=1 [αi(Mi +Ni)− γi(2bi + 1)] > 0. (2.6) In these conditions, the integral operator Hn(z) = ∫ z 0 n∏ i=1 ( fi(t) hi(t) )αi ( g′i(t) )γi dt is in the class K(λ). Proof. We calculate the first and second order derivatives for Hn and we obtain H ′ n(z) = n∏ i=1 ( fi(z) hi(z) )αi ( g ′ i(z) )γi and H ′′ n(z) = n∑ i=1 ( fi(z) gi(z) )αi−1 . (gi(z)) γi−1 × [ f ′ i (z)hi(z)− h ′ i(z)fi(z) h2i (z) .αig ′ i(z) + γi fi(z) hi(z) g ′′ i (z) ] × n∏ k=1 k 6=i ( fk(z) hk(z) )αk ( g ′ k(z) )γk . 5 Nguyen Van Tuan, Adriana Oprea and Daniel Breaz Further we have zH ′′ n(z) H ′n(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) . (2.7) From Eq. (2.7), we get zH ′′ n(z) H ′n(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) ) − n∑ i=1 αi ( zh ′ i(z) hi(z) ) + n∑ i=1 γi ( zg ′′ i (z) g ′ i(z) − zg ′ i(z) gi(z) + 1 ) + n∑ i=1 γi ( zg ′ i(z) gi(z) − 1 ) . (2.8) From Eqs (2.7) and (2.8), we get∣∣∣∣∣H ′′ n(z) H ′n(z) ∣∣∣∣∣ ≤ n∑ i=1 αi ∣∣∣∣∣zf ′ i (z) fi(z) ∣∣∣∣∣+ n∑ i=1 αi ∣∣∣∣∣zh ′ i(z) hi(z) ∣∣∣∣∣+ n∑ i=1 γi ∣∣∣∣∣zg ′′ i (z) g ′ i(z) − g ′ i(z) gi(z) + 1 ∣∣∣∣∣ + n∑ i=1 γi ∣∣∣∣∣zg ′ i(z) gi(z) − 1 ∣∣∣∣∣ . (2.9) Since functions gi ∈ Gbi , 0 ≤ bi < 1, for i = 1, 2, ..., n, using inequality Eq. (1.8) we get∣∣∣∣∣zH ′′ n(z) H ′ n(z) ∣∣∣∣∣ ≤ n∑ i=1 αiMi + n∑ i=1 αiNi + n∑ i=1 γibi ∣∣∣∣∣zg ′ i(z) gi(z) ∣∣∣∣∣+ n∑ i=1 γi ∣∣∣∣∣zg ′ i(z) gi(z) − 1 ∣∣∣∣∣ ≤ n∑ i=1 αi(Mi +Ni) + n∑ i=1 γibi (∣∣∣∣∣zg ′ i(z) gi(z) − 1 ∣∣∣∣∣+ 1 ) + n∑ i=1 γi ∣∣∣∣∣zg ′ i(z) gi(z) − 1 ∣∣∣∣∣ ≤ n∑ i=1 αi(Mi +Ni) + n∑ i=1 2γibi + n∑ i=1 γi = n∑ i=1 [αi(Mi +Ni) + γi(2bi + 1)] = 1− λ. (2.10) So, the integral operator Hn is in the class K(λ). If we consider n = 1, α1 = α, γ1 = γ, f1 = f , h1 = h and g1 = g in Theorem 2.1, we get the following corollary: 6 Some properties of a new integral operator Corollary 2.1. Let f, g, h ∈ A, where g ∈ Gb, 0 < b ≤ 1. For any M,N ≥ 1, which verify the following conditions∣∣∣∣∣zf ′ (z) f(z) ∣∣∣∣∣ ≤M, ∣∣∣∣∣zh ′ (z) h(z) ∣∣∣∣∣ ≤ N, ∣∣∣∣∣zg ′ (z) g(z) − 1 ∣∣∣∣∣ < 1, for all z ∈ U, with real number α, α > 0, with λ = 1 − α(M +N) − γ(2b + 1) > 0. In these conditions, the integral operator H1(z) = ∫ z 0 ( f(t) h(t) )α ( g ′ (t) )γ dt is in the class K(λ). Theorem 2.2. Let fi ∈ S∗(βi) and hi ∈ S∗(δi) with 0 ≤ βi, δi < 1 and gi ∈ K(λi), 0 ≤ λi < 1, for i = 1, 2, ..., n. If αi are real numbers with αi > 0, for i = 1, 2, ...n, so that n∑ i=1 [αi(βi + δi + 2) + γi(1− λi)] < 1. In these conditions, the integral operator Hn(z) = ∫ z 0 n∏ i=1 ( fi(t) hi(t) )αi ( g ′ i(t) )γi dt is convex of order ρ = 1 + ∑n i=1[γi(λi − 1)− αi(βi + δi + 2)], for all i = 1, 2, ..., n. Proof. After the same steps taken in the proof of Theorem 2.1, we get zH ′′ n(z) H ′n(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) . Further, we obtain ∣∣∣∣∣zH ′′ n(z) H ′n(z) ∣∣∣∣∣ = ∣∣∣∣∣ n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) ∣∣∣∣∣ ≤ n∑ i=1 αi ∣∣∣∣∣zf ′ i (z) fi(z) ∣∣∣∣∣+ n∑ i=1 αi ∣∣∣∣∣zh ′ i(z) hi(z) ∣∣∣∣∣+ n∑ i=1 γi ∣∣∣∣∣zg ′′ i (z) g ′ i(z) ∣∣∣∣∣ ≤ n∑ i=1 αi (∣∣∣∣∣zf ′ i (z) fi(z) − 1 ∣∣∣∣∣+ 1 ) + n∑ i=1 αi (∣∣∣∣∣zh ′ i(z) hi(z) − 1 ∣∣∣∣∣+ 1 ) + n∑ i=1 γi ∣∣∣∣∣zg ′′ i (z) g ′ i(z) ∣∣∣∣∣ ≤ n∑ i=1 αi(βi + 1) + n∑ i=1 αi(δi + 1) + n∑ i=1 γi(1− λi) = n∑ i=1 [αi(βi + δi + 2) + γi(1− λi). (2.11) 7 Nguyen Van Tuan, Adriana Oprea and Daniel Breaz From Eq. (2.11), we get∣∣∣∣∣zH ′′ n(z) H ′ n(z) ∣∣∣∣∣ ≤ n∑ i=1 [αi(βi + δi + 2) + γi(1− λi)] = 1− ρ. (2.12) So, the integral operator Hn is convex of order ρ = 1 + ∑n i=1[γi(λi − 1)− αi(βi + δi + 2)], for i = 1, 2, ..., n. If we consider n = 1, α1 = α, γ1 = γ, f1 = f , h1 = h and g1 = g in Theorem 2.2, we get the following corollary. Corollary 2.2. Let f ∈ S∗(β), h ∈ S∗(δ), 0 ≤ β < 1, 0 ≤ δ < 1 and g ∈ K(λ), 0 ≤ λ < 1. If α is a real number, with α > 0 and α(β + δ + 2) + γ(1− λ) < 1. In these conditions, the integral operator H(z) = ∫ z 0 ( f(t) h(t) )α ( g ′ (t) )γ dt is convex of order 1 + γ(λ− 1)− α(β + δ + 2). Theorem 2.3. Let functions fi ∈ SP(α, β), hi ∈ SP(δ, η), with α > 0 and δ > 0, β ∈ [0, 1), η ∈ [0, 1) and gi ∈ N (λi), λi > 1 for i = 1, 2, ..., n. For anyMi, Ni ≥ 1, i = 1, 2, ..., n, which verify∣∣∣∣∣zf ′ i (z) fi(z) ∣∣∣∣∣ ≤Mi, ∣∣∣∣∣zh ′ i(z) hi(z) ∣∣∣∣∣ ≤ Ni for all z ∈ U. There are real numbers αi, with αi > 0 for i = 1, 2, ..., n, so that ρ = 1 + n∑ i=1 [αi(Mi +Ni + 2α− 2η) + γi(λi + 1)] > 1 In these conditions, the integral operator Hn(z) = ∫ z 0 n∏ i=1 ( fi(t) hi(t) )αi ( g ′ i(t) )γi dt is in the class N (ρ). Proof. After the same steps as in the proof of Theorem 2.1, we get: zH ′′ n(z) H ′n(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) . We have 8 Some properties of a new integral operator zH ′′ n(z) H ′n(z) + 1 = n∑ i=1 αi zf ′ i (z) fi(z) − n∑ i=1 αi zh ′ i(z) hi(z) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) + 1 = n∑ i=1 αi ( zf ′ i (z) fi(z) − (α+ β) ) − n∑ i=1 αi ( zh ′ i(z) hi(z) − (δ + η) ) + n∑ i=1 αi(α+ β − δ − η) + n∑ i=1 γi ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1. (2.13) We calculate the real part of both terms in the above expression and obtain Re ( zH ′′ n(z) H ′ n(z) + 1 ) = n∑ i=1 αiRe ( zf ′ i (z) fi(z) − (α+ β) ) − n∑ i=1 αiRe ( zh ′ i(z) hi(z) − (δ + η) ) + n∑ i=1 αi(α+ β − δ − η) + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1 = n∑ i=1 αiRe [( zf ′ i (z) fi(z) − (α+ β) ) − ( zh ′ i(z) hi(z) − (δ + η) )] + n∑ i=1 αi(α + β − δ − η) + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1. (2.14) Since Reω ≤ |ω|, we have Re ( zH ′′ n(z) H ′n(z) + 1 ) ≤ n∑ i=1 αi ∣∣∣∣∣ ( zf ′ i (z) fi(z) − (α+ β) ) − ( zh ′ i(z) hi(z) − (δ + η) )∣∣∣∣∣ + n∑ i=1 αi(α+ β − δ − η) + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1 ≤ n∑ i=1 αi ∣∣∣∣∣zf ′ i (z) fi(z) − (α+ β) ∣∣∣∣∣+ n∑ i=1 αi ∣∣∣∣∣zh ′ i(z) hi(z) − (δ + η) ∣∣∣∣∣ + n∑ i=1 αi(α + β − δ − η) + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1. (2.15) Since fi ∈ SP(α, β), α > 0, β ∈ [0, 1) and hi ∈ SP(δ, η), δ > 0, η ∈ [0, 1), for i = 1, 2, ..., n and gi ∈ N (λi), λi > 1, i = 1, 2, ..., n, we have∣∣∣∣∣zf ′ i (z) fi(z) − (α+ β) ∣∣∣∣∣ ≤ Re ( zf ′ i (z) fi(z) ) + α− β, 9 Nguyen Van Tuan, Adriana Oprea and Daniel Breaz ∣∣∣∣∣zh ′ i(z) hi(z) − (δ + η) ∣∣∣∣∣ ≤ Re ( zh ′ i(z) hi(z) ) + δ − η, and Re ( zg ′′ i (z) g ′ i(z) + 1 ) ≤ λi, λi > 1, i = 1, 2, ...n, z ∈ U. Using above inequalities, we get Re ( zH ′′ n(z) H ′n(z) + 1 ) ≤ n∑ i=1 αi ( Re zf ′ i (z) fi(z) + α− β ) + n∑ i=1 αi ( Re zh ′ i(z) hi(z) + δ − η ) + n∑ i=1 αi(α+ β − δ − η) + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1 ≤ n∑ i=1 αi ∣∣∣∣∣zf ′ i (z) fi(z) ∣∣∣∣∣+ n∑ i=1 αi ∣∣∣∣∣zh ′ i(z) hi(z) ∣∣∣∣∣+ n∑ i=1 αi(α− β + δ − η) + n∑ i=1 αi(α+ β − δ − η) + n∑ i=1 γiRe ( zg ′′ i (z) gi(z) + 1 ) − n∑ i=1 γi + 1 ≤ n∑ i=1 αi(Mi +Ni) + n∑ i=1 αi(2α− 2η) + n∑ i=1 γiλi + n∑ i=1 γi + 1 = n∑ i=1 [αi(Mi +Ni + 2α− 2η) + γi(λi + 1)] + 1. (2.16) So, we obtain Re ( zH ′′ n(z) H ′n(z) + 1 ) ≤ 1 + n∑ i=1 [αi(Mi +Ni + 2α− 2η) + γi(λi + 1)] = ρ. So, the integral operator Hn is in the class N (ρ). If we consider n = 1, α1 = α, γ1 = γ, f1 = f , h1 = h and g1 = g in the Theorem 2.3, we obtain the following corollary. Corollary 2.3. Let functions f ∈ SP(α, β), h ∈ SP(δ, η) with α > 0, δ > 0, β ∈ [0, 1), η ∈ [0, 1) and g ∈ N (λ), λ > 1. For anyM,N ≥ 1, which verify∣∣∣∣∣zf ′ (z) f(z) ∣∣∣∣∣ ≤M, ∣∣∣∣∣zh ′ (z) h(z) ∣∣∣∣∣ ≤ N for all z ∈ U. There is real number α, α > 0, so that ρ = 1 + α(M +N + 2α− 2η) + γ(λ+ 1) > 1 10 Some properties of a new integral operator In these conditions, the integral operator Hn(z) = ∫ z 0 ( f(t) h(t) )α ( g ′ (t) )γ dt is in the class N (ρ). Theorem 2.4. Let fi ∈ S∗λi(b), hi ∈ S∗δi(b), gi ∈ Cλi(b), with 0 ≤ λi < 1, 0 ≤ δi < 1 for i = 1, 2, ..., n and b ∈ C− {0}. Also, let αi be real numbers, with αi > 0 for i = 1, 2, ..., n. If 0 ≤ 1 + n∑ i=1 [αi(λi + δi − 2) + γi(λi − 1) < 1, then the integral operator Hn(z) = ∫ z 0 n∏ i=1 ( fi(t) hi(t) )αi (g,i(t)) γi dt is in the class Cµ(b), with µ = 1 + ∑n i=1[αi(λi + δi − 2) + γi(λi − 1), for i = 1, 2, ..., n. Proof. From previous Theorems, we obtain zH ′′ n(z) H ′ n(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) = n∑ i=1 αi [( zf ′ i (z) fi(z) − 1 ) − ( zh ′ i(z) hi(z) − 1 )] + n∑ i=1 γi zg ′′ i (z) g ′ i(z) . (2.17) Multiplying relation Eq. (2.17) with 1/b, we get 1 b zH ′′ n(z) H ′n(z) = n∑ i=1 αi 1 b [( zf ′ i (z) fi(z) − 1 ) − ( zh ′ i(z) hi(z) − 1 )] + n∑ i=1 γi 1 b zg ′′ i (z) g ′ i(z) = n∑ i=1 αi 1 b ( zf ′ i (z) fi(z) − 1 ) − n∑ i=1 αi 1 b ( zh ′ i(z) hi(z) − 1 ) + n∑ i=1 γi 1 b zg ′′ i (z) g ′ i(z) . (2.18) Further, we have ∣∣∣∣∣1b zH ′′ n(z) H ′n(z) ∣∣∣∣∣ = ∣∣∣∣∣ n∑ i=1 αi 1 b ( zf ′ i (z) fi(z) − 1 ) − n∑ i=1 αi 1 b ( zh ′ i(z) hi(z) − 1 ) + n∑ i=1 γi 1 b zg ′′ i (z) g ′ i(z) ∣∣∣∣∣ ≤ n∑ i=1 αi ∣∣∣∣∣1b ( zf ′ i (z) fi(z) − 1 )∣∣∣∣∣+ n∑ i=1 αi ∣∣∣∣∣1b ( zh ′ i(z) hi(z) − 1 )∣∣∣∣∣+ n∑ i=1 γi ∣∣∣∣∣1b zg ′′ i (z) g ′ i(z) ∣∣∣∣∣ . (2.19) Since fi ∈ S∗λi(b), hi ∈ S∗δi(b) and gi ∈ Cλi(b) for i = 1, 2, ..., n, we have 11 Nguyen Van Tuan, Adriana Oprea and Daniel Breaz ∣∣∣∣∣1b ( zf ′ i (z) f ′ i (z) − 1 )∣∣∣∣∣ ≤ 1− λi, ∣∣∣∣∣1b ( zh ′ i(z) h ′ i(z) − 1 )∣∣∣∣∣ ≤ 1− δiand ∣∣∣∣∣1b zg ′′ i (z) g ′ i(z) ∣∣∣∣∣ ≤ 1− λi. (2.20) So, from Eq. (2.19), we get∣∣∣∣∣1b zH ′′ n(z) H ′n(z) ∣∣∣∣∣ ≤ n∑ i=1 αi(1− λi) + n∑ i=1 αi(1− δi) + n∑ i=1 γi(1− λi) = n∑ i=1 [αi(2− λi − δi) + γi(1− λi) = 1 + µ. (2.21) Since 0 ≤ 1 +∑ni=1[αi(λi + δi − 2) + γi(λi − 1) < 1, we get Hn is in the class Cµ(b), with µ = 1 + ∑n i=1[αi(λi + δi − 2) + γi(λi − 1). If we consider n = 1, α1 = α, γ1 = γ, f1 = f , h1 = h and g1 = g in Theorem 2.4, we get the following corollary. Corollary 2.4. Let f ∈ S∗λ and h ∈ S∗δ , g ∈ Cλ(b) with 0 ≤ λ < 1, 0 ≤ δ < 1 and b ∈ C − {0}. Also, let α be a real number, with α > 0. If 0 ≤ 1+α(λ+ δ−2)+γ(λ−1) < 1, then the integral operator H1(z) = ∫ z 0 ( f(t) h(t) )α ( g ′ (t) )γ dt is in the class Cµ(b), with µ = 1 + α(λ+ δ − 2) + γ(λ− 1). Theorem 2.5. Let fi, gi, hi ∈ A, where gi ∈ N (λi), with λi > 1 for i = 1, 2, ..., n. For any λi > 1, and fi, hi verifying conditions∣∣∣∣∣zf ′ i (z) fi(z) − 1 ∣∣∣∣∣ ≤ 1, ∣∣∣∣∣zh ′ i(z) hi(z) − 1 ∣∣∣∣∣ ≤ 1, (z ∈ U) there are real numbers αi with αi > 0 so that µ = 1 + ∑n i=1(2αi + γiλi − γi) for i = 1, 2, ..., n. In these conditions, the integral operator Hn(z) = ∫ z 0 n∏ i=1 ( fi(t) hi(t) )αi ( g ′ i(t) )γi dt is in the class N (µ). Proof. From the previous Theorems, we obtain zH ′′ n(z) H ′n(z) = n∑ i=1 αi ( zf ′ i (z) fi(z) − zh ′ i(z) hi(z) ) + n∑ i=1 γi zg ′′ i (z) g ′ i(z) = n∑ i=1 αi [( zf ′ i (z) fi(z) − 1 ) − ( zh ′ i(z) hi(z) − 1 )] + n∑ i=1 γi zg ′′ i (z) g ′ i(z) 12 Some properties of a new integral operator = n∑ i=1 αi [( zf ′ i (z) fi(z) − 1 ) − ( zh ′ i(z) hi(z) − 1 )] + n∑ i=1 γi ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi. (2.22) From (2.22), we have Re ( zH ′′ n(z) H ′n(z) + 1 ) = n∑ i=1 αiRe [( zf ′ i (z) fi(z) − 1 ) − ( zh ′ i(z) hi(z) − 1 )] + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1. (2.23) Since gi ∈ N (λi), i = 1, 2, ..., n and Re(ω) ≤ |ω| and applying the conditions from the hypothesis of Theorem 2.5 and Eq. (2.23), we get Re ( zH ′′ n(z) G′n(z) + 1 ) ≤ n∑ i=1 αi ∣∣∣∣∣ ( zf ′ i (z) fi(z) − 1 ) − ( zh ′ i(z) hi(z) − 1 )∣∣∣∣∣ + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1 ≤ n∑ i=1 αi ∣∣∣∣∣ ( zf ′ i (z) fi(z) − 1 )∣∣∣∣∣+ n∑ i=1 αi ∣∣∣∣∣ ( zh ′ i(z) hi(z) − 1 )∣∣∣∣∣ + n∑ i=1 γiRe ( zg ′′ i (z) g ′ i(z) + 1 ) − n∑ i=1 γi + 1 ≤ n∑ i=1 αi + n∑ i=1 αi + n∑ i=1 γiλi − n∑ i=1 γi + 1 = n∑ i=1 (2αi + γiλi − γi) + 1. (2.24) So, Hn is in the class N (µ), with µ = 1 + ∑n i=1(2αi + γiλi − γi), i = 1, 2, ..., n. If consider n = 1, α1 = α, γ1 = γ, f1 = f , h1 = h and g1 = g in Theorem 2.5, we get the following corollary: Corollary 2.5. Let f, h ∈ A, where g ∈ N (λ), λ > 1 and f, h verify conditions:∣∣∣∣∣zf ′ (z) f(z) − 1 ∣∣∣∣∣ ≤ 1, ∣∣∣∣∣zh ′ (z) h(z) − 1 ∣∣∣∣∣ ≤ 1, z ∈ U There is real number α with α > 0 so that µ = 1+2α+γ(λ−1). In these conditions, the integral operator H1(z) = ∫ z o ( f(t) h(t) )α ( g ′ (t) )γ dt is in the class N (µ), with µ = 1 + 2α+ γ(λ− 1). 13 Nguyen Van Tuan, Adriana Oprea and Daniel Breaz 3. Conclusion Our present results were motivated essentially by several recent works dealing with the interesting problem of finding convexity of order for some integral operators (see, for example, [2, 7, 8, 10]). In our study here, we have successfully determined the convexity properties for the functions Hn(z) given by the general family of integral operators in Eq. (2.1). For gi ∈ Gbi , 0 < bi ≤ 1 (i = 1, . . . , n), we study the convexity order of Hn (Theorem 2.1) and the next result find the convexity order ρ of Hn, when fi ∈ S∗(βi), hi ∈ S∗(δi), 0 ≤ βi, δi < 1, for i = 1, . . . , n (Theorem 2.2 ). These theorems are serval corollaries (Corollary 2.1 and corollary 2.2). The following results the convexity of Hn, when fi ∈ SP(α, β), hi ∈ SP(δ, η), α > 0, δ > 0, β ∈ [0, 1), η ∈ [0, 1) and gi ∈ N (λi), λi > 1 and when fi ∈ S∗λi(b), hi ∈ S∗δi(b), gi ∈ Cλi(b), 0 ≤ λi, δi < 1 for i = 1, . . . , n, b ∈ C− {0} (Theorem 2.3 and Theorem 2.4). Also for particular cases we get from these theorems Corollary 2.3 and Corollary 2.4. Also, if gi ∈ N (λi), λi > 1, for i = 1, . . . , n, the last result studies convexity conditions so that Hn ∈ N (µ) (Theorem 2.5 and for a special case Corollary 2.5). REFERENCES [1] D. Breaz, L. Stanciu, 2002. Some properties of a general integral operator. Bulletin of the Transilvania University of Bras¸ov, Series III: Mathematics, Informatics, Physics. [2] D. Breaz, L. Stanciu, 2012. Some properties of a general integral operator. Bulletin of the Transilvania University of Bras¸ov, Series III: Mathematics, Informatics, Physics, Vol. 5 (54), Special Issue: Proceedings of the Seventh Congress of Romanian M