The Convexity of an Integral Product by the means 𝜷-spirallikeness

Lemma 1. 𝑓(𝑧) is 𝛽-spirallike if and only if there is a starlike function 𝑠(𝑧),such that 𝑓(𝑧) 𝑧 = (𝑠(𝑧) 𝑧) 𝛼 (𝑧∈𝑈), Where 𝛼= 𝑒−𝑖𝛽cos𝛽.

Theorem 1. Let 𝐹𝑛(𝑧) be the integral operator given by (1), for 𝑗∈{1,⋅⋅⋅⋅, 𝑛}, if (i) 𝑓𝑗 is 𝛽-spirallike for every 𝑗,(− 𝜋 2 < 𝛽< 𝜋 2). (ii)∑ 1 cos𝛽𝑗 𝑛 𝑗=1 < 𝑛+ 1. (iii) 𝛼𝑗= 𝑒𝑖𝛽𝑗. Then 𝐹𝑛(𝑧) is convex.

Corollary 1. Let 𝑓(𝑧) be 𝛽-spirallike,(− 𝜋 2 < 𝛽< 𝜋 2), if (i)𝛼= 𝑒𝑖𝛽. (ii)cos𝛽> 𝑛 𝑛+1,(𝑛 is a positive integer). Then: ∫𝑡 𝑛((ℜ𝛼)2+ℜ𝛼−1 ℜ𝛼 ) (𝑓(𝑡)

Corollary 2 Let 𝑓(𝑧) be 𝛽-spirallike,(− 𝜋 6 < 𝛽< 𝜋 6), if 𝛼= 𝑒𝑖𝛽 Then: ∫𝑡 (ℜ𝛼)2+ℜ𝛼−1 ℜ𝛼 (𝑓(𝑡) 𝑡) 𝛼 𝑑𝑡 𝑧 0

Theorem 2. Let 𝐺𝑛(𝑧) be the integral operator given by (2), for 𝑗∈{1,⋅⋅⋅⋅, 𝑛}, if (i) 𝑓𝑗 is 𝛽-spirallike for every 𝑗,(− 𝜋 2 < 𝛽< 𝜋 2). (ii)∑ cos𝛽𝑗 𝑛 𝑗=1 < 1. (iii) 𝛼𝑗= 𝑒𝑖𝛽𝑗. Then 𝐺𝑛(𝑧) is convex.

Corollary 3. Let 𝑓(𝑧) be 𝛽-spirallike,(− 𝜋 2 < 𝛽< 𝜋 2), if (i)𝛼= 𝑒𝑖𝛽. (ii)cos𝛽< 1 𝑛,(𝑛 is a positive integer). Then: ∫(𝑓(𝑡) 𝑡) 𝑛𝛼 𝑑𝑡 𝑧

Corollary 4 Let 𝑓(𝑧) be 𝛽-spirallike,(− 𝜋 2 < 𝛽< 𝜋 2), if 𝛼= 𝑒𝑖𝛽 Then: ∫(𝑓(𝑡) 𝑡) 𝛼 𝑑𝑡 𝑧 0

Definition 1. A function 𝑓∈𝑈 is called 𝛽-spirallike if ℜ(𝑒𝑖𝛽𝑧𝑓′(𝑧) 𝑓(𝑧) ) > 0, (𝑧∈𝑈) for a real number 𝛽(− 𝜋 2 < 𝛽< 𝜋 2). Also we denote by 𝐾 the class of convex functions in 𝐴 normalized by 𝑓(0) = 𝑓′(0) −1 = 0, and ℜ{𝑧𝑓′′(𝑧)

Definition 2. Let 𝐹𝑛(𝑧) = ∫𝑡 (ℜ𝛼1)2+ℜ𝛼1−1 ℜ𝛼1 (𝑓1(𝑡) 𝑡 ) 𝛼1 ⋅⋅⋅⋅⋅𝑡 (ℜ𝛼𝑛)2+ℜ𝛼𝑛−1

Definition 3. Let 𝐺𝑛(𝑧) = ∫(𝑓1(𝑡) 𝑡 ) 𝛼1 ⋅⋅⋅⋅⋅(𝑓𝑛(𝑡) 𝑡 ) 𝛼𝑛 𝑑𝑡