β 1.6 We denote this class by Kp β, γ The class Kp β, γ was studied by Aouf We note K β, γ was studied by Libera that K1 β, γ A function f ∈ A p is called quasi-convex of order β type γ, if there exists a function g z ∈ Cp γ such that Re zf z g z z ∈ U, > β, 1.7 ∗ ∗ where ≤ β, γ < p We denote this class by Kp β, γ Clearly f z ∈ Kp β, γ ⇔ zf z /p ∈ Kp β, γ The generalized Srivastava-Attiya operator Js,b f z : A p → A p in is introduced by Js,b f z Gs,b z ∗ f z z ∈ U : b ∈ C \ Z0 {0, −1, −2, −3, }, s ∈ C, p ∈ N 1.8 where Gs,b z φ z, s, b b s φ z, s, b − b−s , zp b bs z1 p b s s ··· 1.9 It is not difficult to see from 1.8 and 1.9 that Js,b f z zp ∞ n 1 n s b b an p zn p 1.10 Journal of Inequalities and Applications When p 1, the operator Js,b is well-known Srivastava-Attiya operator Using the operator Js,b , we now introduce the following classes: S∗ p,s,b γ f z ∈ A p : Js,b f z ∈ S∗ γ p , Cp,s,b γ f z ∈ A p : Js,b f z ∈ Cp γ , 1.11 Kp,s,b β, γ f z ∈ A p : Js,b f z ∈ Kp β, γ , ∗ Kp,s,b β, γ ∗ f z ∈ A p : Js,b f z ∈ Kp β, γ In this paper, we will establish inclusion relation for these classes and investigate SrivastavaAttiya operator for these classes We note that for s σ, b p, we get Jung-Kim-Srivastava for s 1, b c 6, ; p, we get the generalized Libera integral operator 8, ; for s −k being any negative integer, b was studied by S˘ l˘ gean 10 aa 0, and p 1, the operator J−k,0 Dk f z Inclusion Relation In order to prove our main results, we will require the following lemmas Lemma 2.1 see 11 Let w z be regular in U with w on the circle |z| r at a given point z0 ∈ U, then z0 w z0 k ≥ If |w z | attains its maximum value kw z0 , where k is a real number and v1 iv2 , and let ψ u, v be a complex function, Lemma 2.2 see 12 Let u u1 iu2 , v ψ : D → C, D ⊂ C × C Suppose that ψ satisfies the following conditions: i ψ u, v is continuous in D, ii 1, ∈ D and Re{ψ 1, } > 0, iii Re{ψ iu2 , v1 } ≤ for al iu2 , v1 ∈ D such that v1 ≤ − u2 /2 Let h z c1 z c1 z2 · · · be analytic in U, such that h z , zh z Re{ψ h z , zh z } > 0, z ∈ U then Re h z > for z ∈ U ∈ D for z ∈ U If Our first inclusion theorem is stated as follows Theorem 2.3 S∗ γ ⊂ S∗ p,s.b p,s 1,b γ for any complex number s Proof Let f z ∈ S∗ γ , and set p,s,b z Js 1,b f z Js 1,b f z −γ p−γ h z , 2.1 where h z Journal of Inequalities and Applications c1 z z Js · · · Using the identity c2 z2 1,b f p− z b Js 1,b f z b Js,b f z , 2.2 we have Js,b f z Js 1,b f z z Js 1,b f z Js 1,b f z 1 Js,b f z Js 1,b f z b b −p b 2.3 p−γ h z −p γ , b Differentiating 2.3 , logarithmically with respect to z, we obtain z Js,b f z Js,b f z −γ p − γ zh z p−γ h z Now, from the function ψ u, v , by taking u h z, v γ −p p−γ u b 2.4 zh z in 2.4 as p−γ v p−γ u ψ u, v p−γ h z γ −p b , 2.5 it is easy to see that the function ψ u, v satisfies condition i and ii of Lemma 2.2, in D C − { γ − p b / γ − p } × C To verify condition iii , we calculate as follows: Re ψ iu2 , v1 Re Re Re p − γ v1 p − γ iu2 p − γ v1 γ − p p−γ p−γ γ −p p−γ γ −p p−γ 1−p b γ 2 1−p b γ 2.6 v1 b γ −p b 1−p p − γ u2 − i p − γ u2 v1 − i p − γ v1 u2 b b 1−p 2 u2 b b 2 u2 p − γ u2 p−γ ≤− γ −p γ 1 b u2 γ < 0, where v1 ≤ − u2 /2 and iu2 , v1 ∈ D Therefore, the function ψ u, v satisfies the conditions of Lemma 2.2 This shows that if Re h z , zh z > z ∈ U , then Re h z > 0, z∈U 2.7 Journal of Inequalities and Applications if f ∈ S∗ γ , then s ∗ S∗ p,s,b γ ⊂ Sp,s 1,b γ 2.8 This completes the proof of Theorem 2.3 Theorem 2.4 Cp,s,b γ ⊂ Cp,s γ , for any complex number s 1,b Proof Consider the following: z Js,b f z p f z ∈ Cp,s,b γ ⇐⇒ Js,b f z ∈ Cp γ ⇐⇒ zf z p ⇐⇒ Js,b ∈ S∗ γ ⇐⇒ p zf z ∈ S∗ p,s p ⇒ ⇐⇒ z Js p 1,b f 1,b zf z ∈ S∗ p,s,b γ p 1,b ∈ S∗ γ ⇐⇒ Js p z ⇐⇒ f z ∈ Cp,s γ ⇐⇒ Js 1,b ∈ S∗ γ p zf z p 1,b f 2.9 ∈ S∗ γ p z ∈ Cp γ γ , which evidently proves Theorem 2.4 Theorem 2.5 Kp,s,b β, γ ⊂ Kp,s β, γ , for any complex number s 1,b Proof Let f z ∈ Kp,s,b β, γ Then, there exists a function k z ∈ Sp γ such that Re z Js,b f z g z z∈U >β 2.10 Taking the function k z which satisfies Js,b k z g z , we have k z ∈ Sp γ and Re{z Js,b f z /Js,b k z } > β z ∈ U p − β h z , where h z c1 z c2 z2 · · · Now, put z Js 1,b f z / Js 1,b k z − β Using the identity 2.2 we have z Js,b f z Js,b k z Js,b zf z Js,b k z z Js 1,b z Js z Js 1,b z − p− b Js 1,b 1,b k z − p− b Js 1,b k z zf zf z /Js z Js 1,b k 1,b k z zf z − p− /Js 1,b k z b Js 2.11 1,b z − p− zf b z /Js 1,b k z Journal of Inequalities and Applications γ and S∗ γ ⊂ S∗ 1,b γ , we let z Js 1,b k z Since k z ∈ S∗ p,s,b p,s,b p,s γ, where Re H z > z ∈ U thus 2.11 can be written as z Js,b f z Js,b k z z Js 1,b zf /Js z 1,b k z − p− p−γ H z /Js b γ − p− 1,b k p−γ H z z p−β h z β b 2.12 Consider that 1,b f z Js z Js 1,b k p−β h z z β 2.13 Differentiating both sides of 2.13 , and multiplying by z, we have z Js 1,b zf z Js 1,b k z p − β zh z p−β h z β · p−γ H z γ 2.14 2.15 Using 2.14 and 2.12 , we get z Js,b f z Js,b k z Taking u h z ,v −β p − β zh z p−β h z p−γ H z γ − p− b zh z in 2.15 , we form the function ψ u, v as ψ u, v p−β u p−β v p−γ H z γ − p− b 2.16 It is not difficult to see that ψ u, v satisfies the conditions i and ii of Lemma 2.2 in D C × C To verify condition iii , we proceed as follows: p − β v1 p − γ h1 x, y Re ψ iu2 , v1 p − γ h1 x, y γ b −p γ − p− b p − γ h2 x, y , 2.17 where H z h1 x, y ih2 x, y , h1 x, y and h2 x, y being the functions of x and y and Re H z h1 x, y > By putting v1 ≤ − 1/2 u2 , we have Re ψ iu2 , v1 ≤ − p−β u2 p − γ h1 x, y p − γ h1 x, y γ Hence, Re h z > z ∈ U and f z ∈ Kp,s ∗ ∗ Theorem 2.6 Kp,s,b β, γ ⊂ Kp,s 1,b 1,b b −p γ − p− b p − γ h2 x, y < 2.18 β, γ The proof of Theorem 2.5 is complete β, γ for any complex number s Journal of Inequalities and Applications Proof Consider the following: ∗ ∗ f z ∈ Kp,s,b β, γ ⇐⇒ Js,b f z ∈ Kp β, γ ⇐⇒ z Js,b f z p ⇐⇒ Js,b ∈ Kp β, γ zf z p ∈ Kp β, γ zf z ∈ Kp,s ⇒ p ⇐⇒ z Js p ⇐⇒ Js 1,b f 1,b f 1,b ⇒ β, γ ⇐⇒ Js zf z ∈ Kp,s,b β, γ p zf z p 1,b 2.19 ∈ Kp β, γ ∈ Kp β, γ z ∗ z ∈ Kp β, γ ∗ ⇒ f z ∈ Kp,s 1,b β, γ The proof of Theorem 2.6 is complete Integral Operator For c > −1 and f z ∈ A p , we recall here the generalized Bernardi-Libera-Livingston integral operator Lc f z as follows pz c Lc f z zc p tc−1 f t dt ∞ c z c n c 3.1 p p n p n an p z The operator Lc f z when c ∈ N {1, 2, 3, } was studied by Bernardi 13 , for 1, L1 f z was investigated earlier by Libera 14 Now, we have Js,b Lc f z zp ∞ n 1 s b b n c c p p n an p zn p , 3.2 so we get the identity z Js,b Lc f z c p Js,b f z − c Lc f z 3.3 The following theorems deal with the generalized Bernard-Libera-Livingston integral operator Lc f z defined by 3.1 Theorem 3.1 Let c > −γ, ≤ γ < p If f z ∈ S∗ γ , then Lc f z ∈ S∗ γ p,s,b p,s,b Journal of Inequalities and Applications Proof From 3.3 , we have z Js,b Lc f z Js,b Lc f z p Js,b f z c Js,b Lc f z where w z is analytic in U, w −c − 2γ ω z , 1−ω z 3.4 Using 3.3 and 3.4 we get c Js,b f z Js,b Lc f z w z − c − 2γ ω z p p 1−ω z c 3.5 Differentiating 3.5 , we obtain − 2γ w z zw z − 1−w z 1−w z z Js,b f z Js,b f z − c − 2γ zw z p c − c − 2γ w z 3.6 Now we assume that |w z | < z ∈ U Otherwise, there exists a point z0 ∈ U such that kw z0 , k ≥ Putting max |w z | |w z0 | Then by Lemma 2.1, we have z0 w z0 eiθ in 3.6 , we have z z0 and w z0 Re z0 Js,b f z0 Js,b f z0 −γ Re − γ keiθ − eiθ p c − c − 2γ eiθ −2k − γ c 21 c 3.7 γ c − c − 2γ cos θ − c − 2γ ≤ 0, which contradicts the hypothesis that f z ∈ S∗ γ p,s,b Hence, |w z | < 1, for z ∈ U, and it follows 3.4 that Lc f ∈ S∗ γ p,s,b The proof of Theorem 3.1 is complete· Theorem 3.2 Let c > −γ, ≤ γ < p If f ∈ Cp,s,b γ , then Lc f z ∈ Cp,s,b γ Proof Consider the following: f z ∈ Cp,s,b γ ⇐⇒ zf z ∈ S∗ p,s,b γ p ⇒ Lc zf z p ∈ S∗ p,s,b γ ⇐⇒ z Lc f z p ∈ S∗ p,s,b γ ⇐⇒ Lc f z ∈ Cp,s,b γ This completes the proof of Theorem 3.2 Theorem 3.3 Let c > −γ, ≤ γ < p.If f z ∈ Kp,s,b β, γ then Lc f z ∈ Kp,s,b β, γ 3.8 Journal of Inequalities and Applications γ such Proof Let f z ∈ Kp,s,b β, γ Then, by definition, there exists a function g z ∈ S∗ p,s,b that z Js,b f z Js,b g z Re z∈U >β 3.9 Then, z Js,b Lc f z Js,b Lc g z where h z c1 z c2 z2 p−β h z 3.10 · · · From 3.3 and 3.10 , we have Js,b zf z Js,b g z z Js,b f z Js,b g z −β z Js,b Lc zf z cJs,b Lc zf z Js,b Lc g z z Js,b Lc zf z cJs,b Lc g z /Js,b Lc g z z Js,b Lc g z z 3.11 cJs,b Lc zf z /Js,b Lc g z /Js,b Lc g z c γ , then from Theorem 3.1, we have Lc g ∈ S∗ γ Since g z ∈ S∗ p,s,b p,s,b Let z Js,b Lc g z p−γ H z Js,b Lc g z γ, 3.12 where Re H z > z ∈ U Using 3.11 , we have z Js,b f z Js,b g z z Js,b Lc zf z c p−β h z /Js,b Lc g p−γ H z γ c β 3.13 Also, 3.10 can be written as z Js,b Lc f z Js,b Lc g z p−β h z β 3.14 Differentiating both sides, we have z z Js,b Lc f z z Js,b Lc g z p−β h z β p − β zh z Js,b Lc g z , 3.15 or z z Js,b Lc f z Js,b Lc g z z Js,b Lc zf z 3.16 Js,b Lc g z p − β zh z p−β h z β 1−γ H z γ 10 Journal of Inequalities and Applications Now, from 3.13 we have z Js,b f z Js,b g z −β We form the function ψ u, v by taking u ψ u, v p − β zh z p−β h z p−γ H z h z, v γ 3.17 zh z in 3.17 as follows p−β v p−β u c p−γ H z γ c 3.18 It is clear that the function ψ u, v defined in D C × C by 3.18 satisfies conditions i and ii of Lemma 2.2 To verify the condition iii , we proceed as follows: p − β v1 p − γ h1 x, y Re ψ iu2 , v1 p − γ h1 x, y γ c γ c p − γ h2 x, y , 3.19 where H z h1 x, y ih2 x, y , h1 x, y and h2 x, y being the functions of x and y and Re H z h1 x, y > By putting v1 ≤ − 1/2 u2 , we have Re ψ iu2 , v1 ≤ − p−β p − γ h1 x, y u2 p − γ h1 x, y γ c γ c p − γ h2 x, y < 3.20 Hence, Re h z > z ∈ U and Lc f z ∈ Kp,s,b β, γ Thus, we have Lc f z Kp,s,b β, γ The proof of Theorem 3.3 is complete ∈ ∗ ∗ Theorem 3.4 Let c > −γ, ≤ γ < p If f z ∈ Kp,s,b β, γ , then Lc f z ∈ Kp,s,b β, γ Proof Consider the following: ∗ f z ∈ Kp,s,b β, γ ⇐⇒ zf z ∈ Kp,s,,b β, γ ⇒ Lc zf z ⇐⇒ z Lc f z ∈ Kp,s,b β, γ ∈ Kp,s,b β, γ ∗ ⇐⇒ Lc f z ∈ Kp,s,b β, γ , and the proof of Theorem 3.4 is complete Acknowledgement The authors would like to thank the referees of the paper for their helpful suggestions 3.21 Journal of Inequalities and Applications 11 References A W Goodman, “On the Schwarz-Christoffel transformation and p-valent functions,” Transactions of the American Mathematical Society, vol 68, pp 204–223, 1950 M K Aouf, “On a class of p-valent close-to-convex functions of order β and type α,” International Journal of Mathematics and Mathematical Sciences, vol 11, no 2, pp 259–266, 1988 R J Libera, “Some radius of convexity problems,” Duke Mathematical Journal, vol 31, no 1, pp 143– 158, 1964 J.-L Liu, “Subordinations for certain multivalent analytic functions associated with the generalized Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol 19, no 11-12, pp 893–901, 2008 H M Srivastava and A A Attiya, “An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination,” Integral Transforms and Special Functions, vol 18, no 3-4, pp 207–216, 2007 J.-L Liu, “Notes on Jung-Kim-Srivastava integral 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Journal of Mathematical Analysis and Applications, vol 65, no 2, pp 289–305, 1978 13 S D Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol 135, pp 429–446, 1969 14 R J Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol 16, pp 755–758, 1965 ... one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol 176, no 1, pp 138–147, 1993 J.-L Liu, ? ?Some applications of certain integral operator, ” Kyungpook... difficult to see from 1.8 and 1.9 that Js,b f z zp ∞ n 1 n s b b an p zn p 1.10 Journal of Inequalities and Applications When p 1, the operator Js,b is well-known Srivastava-Attiya operator Using... Journal of Inequalities and Applications A function f ∈ A p is said to be in the class Cp α of p-valent convex functions of order α if it satisfies zf z f z Re ≤ α < p, z ∈ U >α The class of p-valent