A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions

Theorem 2.1. Let the function f = h + ¯g be such that h and g are given by (1.1). Furthermore, let (2.1) ∞ X n=2 pn −αun 1 −α |An| + ∞ X n=1 qn −(−1)j−iαvn 1 −α |Bn| ≤1

Theorem 2.1 gives a sufficient condition for the harmonic function φ1 + φ2 to be in the class H(Φi, Ψj; α) where φ1(z) ≡φ1(a1, b1, c1; z) and φ2(z) ≡φ2(a2, b2, c2; z) are the hypergeometric functions defined by (2.3) φ1(z) := zF(a1, b1, c1; z) and φ2(z) := F(a2, b2, c2; z) −1.

Corollary 2.3. Let ak, bk, ck > 0 for k = 1, 2. Furthermore, let (2.4) ∞ X n=2 pn −αun 1 −α (a1)n−1(b1)n−1 (c1)n−1(1)n−1 + ∞ X n=1 qn −(−1)j−iαvn 1 −α

Corollary 2.4. Suppose that ak, bk, ck > 0 for k = 1, 2 and (2.6) ∞ X n=2 pn −αun 1 −α (a1)n−1(b1)n−1 (c1)n−1(1)n + ∞ X n=2 qn −(−1)j−iαvn 1 −α

Theorem 2.5. Let the function f = h + ¯g be such that h and g are given by (1.2). Then f ∈T H(Φi, Ψj; α) if and only if (2.7) ∞ X n=2 pn −αun 1 −α An + ∞ X n=1 qn −(−1)j−iαvn 1 −α Bn ≤1

Theorem 2.5 immediately yields the following three corollaries.

Corollary 2.6. For f = h + ¯g ∈T H(Φi, Ψj; α) where h and g are given by (1.2), we have An ≤ 1 −α pn −αun (n = 2, 3, . . .) and Bn ≤ 1 −α qn −(−1)j−iαvn (n = 1, 2, . . .); the result being sharp, for each n.

Corollary 2.7. Let ak, bk, ck > 0 for k = 1, 2 and φ1, φ2 be given by (2.3). Then a necessary and sufficient condition for the harmonic function Φ(z) = 2z −φ1(z) + φ2(z) to be in the class T H(Φi, Ψj; α) is that (2.4) is satisfied.

Corollary 2.8. If ak, bk, ck > 0 for k = 1, 2, then Ψ(z) = 2z −ψ1(z) + ψ2(z) ∈ T H(Φi, Ψj; α) if and only if condition (2.6) holds, where ψ1, ψ2 are given by (2.5). Note that [4, Theorem 2.6] is a particular case of Corollary 2.7. By making use of Theorem 2.5, we obtain the following growth estimate for functions in the class T H(Φi, Ψj; α).

Theorem 2.9. Let f ∈T H(Φi, Ψj; α), σn = pn −αun (n = 2, 3, . . .) and Γn = qn −(−1)j−iαvn (n = 1, 2, . . .). If {σn} and {Γn} are non-decreasing sequences, then |f(z)| ≤(1 + B1)|z| + 1 −α η  1 −q1 −(−1)j−iαv1 1 −α B1  |z|2, and |f(z)| ≥(1 −B1)|z| −1 −α η 

Corollary 2.10. Under the hypothesis of Theorem 2.9, we have  w ∈C : |w| < 1 η (η −1 + α + (q1 −(−1)j−iαv1 −η)B1)  ⊂f(D). Using Theorem 2.5 it is easily seen that the class T H(Φi, Ψj; α) is convex and closed with respect to the topology of locally uniform convergence so that the closed convex hull of T H(Φi, Ψj; α) equals itself. The next theorem determines the extreme points of T H(Φi, Ψj; α).

Theorem 2.11. Suppose that 0 ≤α < 1, i, j ∈{0, 1}, pn > un ≥0 (n = 2, 3, . . .) and qn > vn ≥0 (n = 1, 2, . . .). Set h1(z) = z, hn(z) = z − 1 −α pn −αun zn (n = 2, 3, . . .) and gn(z) = z + 1 −α qn −(−1)j−iαvn ¯zn (n = 1, 2, . . .). Then f ∈T H(Φi, Ψj; α) if and only if it can be expressed in the form (2.8) f(z) = ∞

Theorem 2.12. Suppose that f, F ∈T H are given by (2.9) with A′ n ≤1 and B′ n ≤1. If f ∈T H(Φi, Ψj; α) then fˆ∗F ∈T H(Φi, Ψj; α).

Corollary 2.13. If f, F ∈T H(Φi, Ψj; α) with pn ≥1 (n = 2, 3, . . .) and qn ≥1 (n = 1, 2, . . .) then fˆ∗F ∈T H(Φi, Ψj; α).

Theorem 2.15. Let f ∈T H(Φi, Ψj; α) and Φ(z) = 2z −φ1(z) + φ2(z); φ1 and φ2 being given by (2.3). If ak, bk > 0, ck > ak + bk for k = 1, 2 and if F(a1, b1, c1; 1) + F(a2, b2, c2; 1) ≤3, then fˆ∗Φ ∈T H(Φi, Ψj; α).

Theorem 2.5 now gives the desired result. □

Theorem 2.16. Let f ∈T H(Φi, Ψj; α), ak, bk > 0 and ck > ak + bk for k = 1, 2. Furthermore, if F(a1, b1, c1; 1) + F(a2, b2, c2; 1) ≤4, then fˆ∗Ψ ∈T H(Φi, Ψj; α) where Ψ(z) = 2z −ψ1(z)+ψ2(z); ψ1 and ψ2 being given by (2.5).

Theorem 2.17. The class T H(Φi, Ψj; α) is closed under convex combinations.

Theorem 3.1. Let the function f = h + ¯g be such that h and g are given by (1.2) and 0 ≤α < 1. Then f ∈T UH(α) if and only if ∞ X n=2 An 1 −α + ∞ X n=1 Bn 1 −α ≤1. Furthermore, if f ∈T UH(α) then An ≤1 −α (n = 2, 3, . . .), Bn ≤1 −α (n = 1, 2, . . .) and (3.1)

Theorem 3.2. Let ak, bk > 0, ck > ak + bk for k = 1, 2. Then a necessary and sufficient condition for the harmonic function Φ(z) = 2z −φ1(z) + φ2(z) to be in the class T UH(α) is that F(a1, b1, c1; 1) + F(a2, b2, c2; 1) ≤3 −α, where φ1 and φ2 are given by (2.3). The upper bound given in (3.1) for f ∈T UH(α) is sharp and equality occurs for the function f(z) = z + B1¯z + (1 −α −B1)¯z2 for B1 ≤1 −α. In a similar fashion, comparable results to Corollary 2.8 and Theorems 2.15, 2.16 for the class T UH(

Lemma 3.3. Let f = h + ¯g ∈H where h and g are given by (1.1) with B1 = g′(0) = 0. Suppose that λ ∈(0, 1]. (i) If P∞ n=2(|An| + |Bn|) ≤λ then f ∈UH(1 −λ); (ii) If P∞ n=2 n(|An| + |Bn|) ≤λ then f ∈UH(1 −λ/2) and is starlike of order 2(1 −λ)/(2 + λ). The results are sharp.

Corollary 3.4. The class T UH(α) is closed under the product ˆ∗. In fact T U H(α)ˆ∗T U H(β) ⊂T U H(1 −(1 −α)(1 −β)) for α, β ∈[0, 1).

Theorem 3.5. For 0 ≤α < 1, the following sharp inclusions hold: (3.2) T K0 H(α) ⊂T U0 H  3 −α 2(2 −α)  ; and (3.3) T S∗0 H (α) ⊂T U 0 H 

Theorem 3.7. The radius of univalence of the class T U0 H(α) is 1/(2(1−α)). This bound is also the radius of starlikeness of T U0 H(α). The radius of convexity of the class T U 0 H(α) is 1/(4(1 −α)).

Theorem 1, p. 284], f is univalent and starlike in |z| < 1/(2(1 −α)). Regarding the radius of convexity, note that ∞ X n=2 n2(An + Bn)rn−1 ≤ ∞ X n=2  An 1 −α + Bn 1 −α  ≤1