Construction of subclasses of univalent harmonic mappings

Lemma 2.1. A sense-preserving harmonic mapping f = h + ¯g is stable univalent (resp. stable starlike, stable convex and stable close-to-convex) if and only if the analytic func- tions Fλ = h+ λg are univalent (resp. starlike, convex and close-to-convex) in D for each |λ| = 1. Let SS0 H, SS∗0 H , SK0 H and SC0 H be subclasses of S0 H consisting of stable univalent, stable starlike, stable convex and stable close-to-convex mappings respectively. Then S∗⊂SS∗0 H ⊂S∗0 H , K ⊂SK0

Theorem 2.2. Suppose that G ⊂S and G ⊲G0 H. Then (i) G0 H ⊂SS0 H; (ii) If f ∈S ∩G0 H, then f ∈G; (iii) If f = h+¯g ∈G0 H, then the harmonic mappings fλ = h+λ¯g ∈G0 H for each |λ| = 1; (iv) If J ⊂G, then J 0 H ⊂G0 H where J 0 H is the harmonic analogue of J .

Theorem 2.2(ii) conveys that every analytic univalent function in G0 H is a member of G. Since the members of G0 H are stable univalent by Theorem 2.2(i), we have the following corollary which follows by [13, Theorem 4, p. 17].

Corollary 2.3. Suppose that G ⊂S and G ⊲G0 H. If f = h + ¯g ∈G0 H, then the analytic mappings Fµ = h + µg are univalent in D for each |µ| ≤1. In particular, h is univalent. Recall that convexity and starlikeness are hereditary properties for conformal mappings which do not extend to harmonic mappings (see [9]). Chuaqui, Duren and Osgood [5] introduced the notion of fully starlike and fully convex functions that do inherit the properties of starlikeness and convexity respectively (see also [20]).

Corollary 2.4. Suppose that G ⊂S and G ⊲G0 H. If G ⊂S∗(resp. G ⊂K), then members of G0 H are fully starlike (resp. fully convex) in D. It is easy to see that if I and J are subclasses of S with I ⊲I0 H and J ⊲J 0 H, then I ∩J ⊲I0 H ∩J 0 H and I ∪J ⊲I0 H ∪J 0 H. The next theorem determines the coefficient bounds for functions in the harmonic analogue G0 H.

Theorem 2.5. Suppose that G ⊂S and G ⊲G0 H. Let the Taylor coefficients an(f) of the series of each f ∈G satisfies |an(f)| ≤p(n) for n = 2, 3, . . . where p is a function of n. Then (a) The respective Taylor coefficients An(f) and Bn(f) of the series of h and g of each function f = h + ¯g ∈G0 H satisfies ||An(f)| −|Bn(f)|| ≤p(n), n = 2, 3, . . . . (b) Let h0 ∈G be such that its Taylor coefficients satisfy |an(h0)| = p(n) for n = 2, 3, . . .. Then for an analytic function g0, the harmonic function f0 = h

Theorem 2.6. Suppose that G ⊂S and G ⊲G0 H. If P(|z|) ≤|f ′(z)| ≤Q(|z|), z ∈D for each f ∈G where P and Q are integrable functions of |z|, then each f ∈G0 H satisfies Z |z| 0 P(ρ) dρ ≤|f(z)| ≤ Z |z| 0 Q(ρ) dρ, z ∈D. In particular, the range of every function f ∈G0 H contains the disk

Theorem 2.7. Suppose that G ⊂S and G ⊲G0 H. Then G is compact if and only if G0 H is compact.

Theorem 2.2(ii), f ∈G. This completes the proof. □ The next theorem investigates the relation between the radius of starlikeness, convexity and close-to-convexity of the classes G and G0 H.

Theorem 2.8. Suppose that G ⊂S and G ⊲G0 H. Then the classes G and G0 H have the same radius of starlikeness, convexity and close-to-convexity.

Theorem 2.9. Suppose that G ⊂S is closed under convolution and G ⊲G0 H. Then (i) The convolution of each member of G0 H with itself is again a member of G0 H; (ii) If (f + g)/2 ∈G for all f, g ∈G, then G0 H is closed under convolution.

Theorem 2.9 immediately gives

Corollary 2.10. Suppose that G ⊂S is a convex set and is closed under convolution. If G ⊲G0 H, then G0 H is closed under convolution. In [10], Goodloe considered the Hadamard product ˜∗of a harmonic function with an analytic function defined as follows: f˜∗ϕ = ϕ˜∗f = h ∗ϕ + g ∗ϕ, where f = h + ¯g is harmonic and ϕ is analytic in D. The next theorem investigates the properties of the product ˜∗.

Theorem 2.11. Suppose that G ⊂S and G ⊲G0 H. Let O be a subfamily of A such that G is closed under convolution with members of O. Then ϕ˜∗f ∈G0 H for all ϕ ∈O and f ∈G0 H.

Theorem 2.12. Suppose that G ⊂S and G ⊲G0 H. Then G is closed under convex combi- nations if and only if G0 H is closed under convex combinations.

Theorem 2.12 immediately yields

Corollary 2.13. Suppose that G ⊂S and G ⊲G0 H. Then G is a convex set if and only if G0 H is a convex set.

Theorem 2.15. A subfamily G0 H ⊂S0 H is a harmonic analogue of some family G ⊂S if and only if G0 H ⊂SS0 H.

Theorem 2.16. Let f = h + ¯g ∈S0 H where h and g are given by (1.1). (i) (Coefficient estimates) If f ∈SS0 H, SS∗0 H or SC0 H, then the sharp inequality ||an|− |bn|| ≤n holds for n = 2, 3, . . .. Equality occurs for the analytic Keebe function k(z) = z/(1 −z)2. In case, f ∈SK0 H then ||an| −|bn|| ≤1 for n = 2, 3, . . ., with the equality occurring for the analytic half-plane mapping l(z) = z/(1 −z). (ii) (Growth estimates and covering theorem) If f ∈SS0 H, SS∗0 H or SC0 H, then we have |z|

Theorem 2.8. Since K ∗S∗⊂S∗, K ∗K ⊂K and K ∗C ⊂C, the convolution properties are easy to deduce from Theorems 2.9(i) and 2.11. □ We close this section with the following remark.

Lemma 3.1. Let I and J be subfamilies of S such that I ∗I ⊂J . If I0 H and J 0 H denote the harmonic analogues of I and J respectively, then (a) If f ∈I0 H, then f ∗f ∈J 0 H; (b) If (f + g)/2 ∈J for all f, g ∈J , then I0 H ∗I0 H ⊂J 0 H.

Lemma 3.2. Suppose that I and J be subfamilies of S. Let O ⊂A be such that f ∗g ∈J for all f ∈I and g ∈O. Then ϕ˜∗f ∈J 0 H for all ϕ ∈O and f ∈I0 H, where I ⊲I0 H and J ⊲J 0 H. Denote by R the class consisting of functions f ∈A which satisfy Re f ′(z) > 0 for z ∈D. By well-known Noshiro-Warschawski Theorem (see [12, Chapter 7, p. 88]), R ⊂S. In [15], MacGregor investigated the properties of functions in the class R. Also,

Theorem 3.3. The class R0 H is the harmonic analogue of R where R0 H = {f = h + ¯g ∈H : Re h′(z) > |g′(z)| for all z ∈D}. In particular, R0 H ⊂SC0 H. Moreover, we have (i) If f = h + ¯g ∈R0 H where h and g are given by (1.1), then ||an| −|bn|| ≤2/n for n = 2, 3, . . .. Equality holds for the function f given by (3.1). (ii) Every function f ∈R0 H satisfies −|z| + 2 log(1 + |z|) ≤|f(z)| ≤−|z| −2 log(1 −|z|), z ∈D and hence the range of each function f ∈R0

Lemma 3.4. If f ∈W is given by (2.1), then |an| ≤2/n2 for n = 2, 3, . . . and −1 + 2 |z| log(1 + |z|) ≤|f ′(z)| ≤−1 −2 |z| log(1 −|z|), z ∈D. The function (3.2) f(z) = −z −2 Z |z| 0 1 t log(1 −t) dt = z + ∞ X n=2

Theorem 3.5. Let f = h+¯g ∈W0 H where h and g are given by (1.1). Then ||an|−|bn|| ≤ 2/n2 for n = 2, 3, . . . and −|z| + 2 Z |z| 0 1 t log(1 + t) dt ≤|f(z)| ≤−|z| −2 Z |z| 0 1 t log(1 −t) dt, z ∈D. In particular, the range f(D) contains the disk |w| < π2/6 −1. All these results are sharp for the function f given by (3.2). Moreover, the following statements regarding the class

Theorem 3.7. The harmonic analogues of the classes U and V are given by U0 H = ( f(z) = z + ∞ X n=2 anzn + ∞ X n=2 bnzn ∈H : ∞ X

Corollary 3.8. Let f = h + ¯g ∈S0 H where h and g are given by (1.1).

Theorem 3.9. The classes U0 H and V0 H are closed under convolutions. Moreover, we have (i) U0 H ∗U0 H ⊂SK0 H; (ii) If ϕ ∈K and f ∈U0 H, then ϕ˜∗f ∈U0 H; (iii) If ϕ ∈K and f ∈V0 H, then ϕ˜∗f ∈V0 H.

Theorem 3.10. SR is the harmonic analogue of SR itself.

Theorem 4.2. Let I and J be subfamilies of S such that Λ[I] ⊂J . Then Λ+ H[I0 H] ⊂J 0 H where I ⊲I0 H and J ⊲J 0 H.

Theorem 4.2 gives the following two corollaries.

Corollary 4.3. Λ+ H[R0 H] ⊂SS∗0 H and Λ+ H[U0 H] ⊂SK0 H

Corollary 4.4. The classes R0 H, W0 H, U0 H and V0 H are preserved under Λ+ H. The Alexander operator Λ provides a one-to-one correspondence between the classes S∗and K: f ∈S∗if and only if Λ[f] ∈K. A similar result holds for the positive harmonic Alexander operator which provides a one-to-one correspondence between the classes SS∗0 H and SK0 H.

Corollary 4.5. f ∈SS∗0 H if and only if Λ+ H[f] ∈SK0 H. However, the inclusion Λ+ H[S∗0 H ] ⊂K0 H is not valid in general. To see this, note that the harmonic Koebe function K given by (2.2) belongs to S∗0 H and Λ+ H[K](z) = 1 6 z(5 −3z) (1 −z)2 −log(1 −z)

Theorem 4.7. Let rC denotes the radius of convexity of the class R0 H(G) for G ∈A. (i) If G ∈S, then rC = 3 −2 √ 2; (ii) If G ∈S∗, then rC = 3 −2 √ 2; (iii) If G ∈K, then rC = 2 − √ 3; (iv) If G ∈R, then rC = √ 5 −2; (v) If G ∈A with Re G′(z) > 1/2, then rC = 3 −2

Theorem 4.8. Suppose that rC denotes the radius of convexity of the class F 0 H(G) for G ∈A. (a) If G ∈S, then rC = 1/5; (b) If G ∈S∗, then rC = 1/5; (c) If G ∈K, then rC = 1/3; (d) If G ∈R, then rC = ( √ 17 −3)/4; (e) If G ∈A with Re G′(z) > 1/2, then rC is the smallest positive root of the equation r4 + 2r3 + 13r2 + 4r −4 = 0. Moreover, all these results are sharp.