Study on new integral operators defined using confluent hypergeometric function

Theorem 2.1 Let 1F1(a,c;z) be given by (1) with B′(0) = a c · γ γ +1 ̸= 0, γ > 0, and Re z1F′ 1(a,c;z) 1F1(a,c;z) > 0, z ∈U i.e. 1F1(a,c;z) ∈S∗. (4) Then the Kummer–Bernardi integral operator given in (2) is a starlike function and B

Corollary 2.1 Let 1F1(a,c;z) be given by (1) with 1F1(a,c;z) ̸= 0 and Re z1F′ 1(a,c;z) 1F1(a,c;z) > 0, z ∈U i.e. 1F1(a,c;z) ∈S∗. Then Kummer–Libera integral operator given in (3) is a starlike function and L[S∗] ⊂S∗ or B(U) ⊂{z ∈C : z = x + iy,x > 0,y ∈R}. Using Theorem 4.4.4 [12, p. 76] and Theorem 2.1, we prove in the next theorem the property that Kummer–Bernardi and Kummer–Libera integral operators have of extending stralike- ness of order 1 2 to stralikeness.

Corollary 2.2 Let 1F1(a,c;z) be given by (1) with 1F1(a,c;z) ̸= 0 and Re z1F′ 1(a,c;z) 1F1(a,c;z) > 1 2, z ∈U i.e. 1F1(a,c;z) ∈S∗ 1 2  . Then the Kummer–Bernardi integral operator given in (2) is a starlike function and B  S∗ 1

Corollary 2.3 Let 1F1(a,c;z) be given by (1) with 1F1(a,c;z) ̸= 0 and Re z1F′ 1(a,c;z) 1F1(a,c;z) > 1 2, z ∈U i.e. 1F1(a,c;z) ∈S∗ 1 2  . Then the Libera–Bernardi integral operator given in (3) is a starlike function and L  S∗ 1

Theorem 2.2 Let 1F1(a,c;z) be given by (1) with 1F′ 1(a,c;0) ̸= 0, γ > 0, and Re z1F′′ 1 (a,c;z) 1F′ 1(a,c;z) + 1  > 0, z ∈U i.e. 1F1(a,c;z) ∈K. (15)

Corollary 2.4 Let 1F1(a,c;z) be given by (1) with 1F′ 1(a,c;0) ̸= 0, γ > 0 and Re z1F′′ 1 (a,c;z) 1F′ 1(a,c;z) + 1  > 0, z ∈U i.e. 1F1(a,c;z) ∈K. Then the Kummer–Libera integral operator given in (3) is convex in U and L[K] ⊂K or L(U) ⊂{z ∈C : z = x + iy,x > 0,y ∈R}.

Corollary 2.5 Let Re[ zB′′(z) B′(z) + 1] > 0. Using the Marx–Strohhäcker result [11, p. 55], we get that Re[ zB′(z) B(z) + 1] > 1 2 i.e. B ∈S∗( 1 2) and B(K) ⊂S∗( 1 2) or B(U) ⊂  z ∈C : z = x + iy,x > 1 2,y ∈R  . For γ = 1, we obtain the following corollary for Kummer–Libera integral operator.

Corollary 2.6 If Re[ zL′′(z) L′(z) + 1] > 0, using the Marx–Strohhäcker result [11, p. 55], we get that Re[ zL′(z) L(z) + 1] > 1 2 i.e. L ∈S∗( 1 2) and L(K) ⊂S∗( 1 2) or L(U) ⊂  z ∈C : z = x + iy,x > 1 2,y ∈R  . The study is concluded with an example of how the results presented in the paper are useful.