New univalent conditions for a family of integral operators

In this paper, we obtain new sufficient conditions for the operator G n , α ( z ) = { [ n ( α − 1 ) + 1 ] ∫ 0 z ( g 1 ( t ) ) α − 1 ⋯ ( g n ( t ) ) α − 1 d t } 1 n ( α − 1 ) + 1 to be univalent in the open unit disc U , where each of the functions g 1 , g 2 , … , g n satisfy the condition | z g i ′ ( z ) g i ( z ) − a | < b , ( i = 1 , 2 , … , n ; z ∈ U ; | a − 1 | < b ≤ a ) .