Fully starlike and fully convex harmonic mappings of order α

Theorem 3]). The next theorem provides a sufficient condition for a sense- preserving harmonic mapping to be fully convex.

Theorem 2.3. A sense-preserving harmonic function f = h + ¯g is fully convex in D if the analytic functions h + ϵg are convex in D for all |ϵ| = 1.

Lemma 2.4 ([8]). Let f = h + ¯g, where h and g are given by (1.1), and let 0 ≤α < 1. If ∞ X n=2 n(n −α) 1 −α |an| + ∞ X n=1 n(n + α) 1 −α |bn| ≤1, then f ∈FKH(α).

Theorem 2.5. Let f = h+¯g ∈H be sense-preserving and let 0 ≤α < 1. Then f ∈FKH(α) if and only if (2.3) |zh′(z)|2  Re  1 + zh′′(z) h′(z)  −α  −|zg′(z)|2  Re

Theorem 2.7. A sense-preserving harmonic function f = h + ¯g is fully starlike in D if the analytic functions h+ϵg are starlike in D for each |ϵ| = 1. Note that the exact radius of full starlikeness of order α (0 ≤α < 1) for the subclasses S∗ H, KH and CH in SH is still unknown. The results in this direction are investigated in the next section. However, if α = 0 then

Theorem 2.7 immediately gives

Corollary 2.8. Suppose that f = h + ¯g ∈SH. (i) If f ∈KH then f is fully starlike in at least |z| < 4 √ 2 −5. (ii) If f ∈CH then f is fully starlike in at least |z| < 3 − √ 8. (iii) If f ∈S∗ H then f is fully starlike in at least |z| < √ 2 −1.

Lemma 2.9 ([9]). Let f = h + ¯g, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n −α 1 −α |an| + ∞ X n=1 n + α 1 −α |bn| ≤1 and 0 ≤α < 1. Then f ∈FS∗ H(α). Corresponding to Theorem 2.5, the analytic characterization of functions

Theorem 2.10. Let f = h+¯g ∈H be sense-preserving and let 0 ≤α < 1. Then f ∈FS∗ H(α) if and only if f(z) ̸= 0 for 0 < |z| < 1 and (2.4) |h(z)|2  Re zh′(z) h(z) −α  −|g(z)|2  Re zg′(z) g(z) + α  > Re(zh(z)g′(z) + 2αh(z)g(z) −zh′(z)g(z))

Theorem 2.13. Let h, g, H and G be analytic functions in the unit disc D, related by zH′(z) = h(z) and zG′(z) = −g(z). Then f = h + ¯g is fully starlike of order α if and only if F = H + ¯G is fully convex of order α, where 0 ≤α < 1. This theorem provides abundant examples of fully convex and fully star- like mappings of order α (0 ≤α < 1). For instance, since the functions fn defined in Example 2.6 are fully starlike in D, the functions Fn(z) = z −[(1 −α)/(n(n + α))]¯zn (z ∈D) are fully convex o

Corollary 2.15. If f = h + ¯g ∈FS∗ H(α) (0 ≤α < 1) and if H and G are the analytic functions defined by zH′(z) = h(z), zG′(z) = −g(z), and H(0) = G(0) = 0, then F = H + ¯G ∈FKH(α).

Theorem 3.1. Let h and g have the form (1.1) with b1 = g′(0) = 0, 0 ≤α < 1 and the coefficients of the series satisfying the conditions (1.4). Then f = h+ ¯g is univalent and fully starlike of order α in the disk |z| < rS, where rS = rS(α) is the real root of the equation (3.2) 2(1 −α)(1 −r)4 + α(1 −r)2 −(1 + r)2 = 0 in the interval (0, 1). Moreover, this result is sharp for each α ∈[0, 1).

Corollary 3.2. Let f ∈S∗0 H (resp. C0 H) and 0 ≤α < 1. Then f is fully starlike of order α in at least |z| < rS, where rS is the real root of (3.2) in (0, 1).

Corollary 2.8 shows that the result in Corollary 3.2 is not sharp if α = 0. Proceeding as in Theorem 3.1 and invoking Lemma 2.4 instead of Lemma 2.9, we have the following result.

Theorem 3.3. Under the hypothesis of Theorem 3.1, f = h + g is uni- valent and fully convex of order α in the disk |z| < rC, where rC = rC(α) is the real root of the equation (3.5) 2(1 −α)(1 −r)5 + α(1 + r)(1 −r)2 −(1 + r)(r2 + 6r + 1) = 0 in the interval (0, 1). In particular, f is univalent and fully convex in |z| < rC(0) ≈0.0614313. The bound rC given by (3.5) is sharp by considering the function f0(z) = 2z −K(z) where K is given by (1.5). In fact, as f0 has real coefficients, we obtain, when θ

Theorem 3.3 immediately gives

Corollary 3.4. Let f ∈S∗0 H (resp. C0 H) and 0 ≤α < 1. Then f is fully convex of order α in at least |z| < rC, where rC is the real root of (3.5) in (0, 1). It is clear that the result in Corollary 3.4 is not sharp if α = 0. Corre- sponding to Theorem 3.1, the next theorem determines the radius of univa- lence and full starlikeness of order α for functions f = h + ¯g ∈H, where the Taylor coefficients of the series of h and g satisfy (1.6).

Theorem 3.5. Let h and g have the form (1.1) with b1 = g′(0) = 0, 0 ≤α < 1 and the coefficients of the series satisfying the conditions (1.6).

Corollary 3.6. Let f ∈K0 H and 0 ≤α < 1. Then f is fully starlike of order α in at least |z| < rS, where rS is the real root of (3.6) in (0, 1). If α = 0 then the result in Corollary 3.6 is not sharp in view of Corollary 2.8. It is expected that Corollary 3.6 can be further improved, and since the harmonic half-plane mapping L given by (1.2) is extremal in K0 H, Example 2.11 motivates the following conjecture: Conjecture A. If f ∈K0 H, then f is fully starlike of order α (0 ≤α < 1) in |z| < rS w

Theorem 3.7. Under the hypothesis of Theorem 3.5, f = h + ¯g is uni- valent and fully convex of order α in the disk |z| < rC, where rC = rC(α) is the real root of the equation (3.7) 2(1 −α)(1 −r)4 + α(1 −r)2 −(r2 + 4r + 1) = 0 in the interval (0, 1). In particular, f is univalent and fully convex in |z| < rC(0) ≈0.0903331. The radius bound rC given by (3.7) is sharp for each α ∈[0, 1) by con- sidering the function f0(z) = h0(z) + g0(z), where h0(z) = 2z −M(z) and g0(z) = N(z) (z ∈D), L = M + N b

Corollary 3.8. If f ∈K0 H and 0 ≤α < 1, then f is fully convex of order α in |z| < rC, where rC is the real root of (3.7). It is known that the result given in Corollary 3.8 is not sharp if α = 0. Since the harmonic half-plane mapping L given by (1.2) gives the sharp bound for α = 0, Example 2.11 motivates the following conjecture: Conjecture B. If f ∈K0 H, then f is fully convex of order α (0 ≤α < 1) in |z| < rS where rC = rC(α) is the positive root of the equation p(r, u0) = 0

Theorem 4.2. Let h and g have the form (1.1) with b1 = g′(0) = 0, 0 ≤α < 1 and the coefficients of the series satisfying the conditions |an| ≤ n + 1 2 2 and |bn| ≤ n −1 2 2 for all n ≥2. Then f = h+ ¯g is univalent and fully starlike of order α in the disk |z| < r0, where r0 = r0(α) is the real root of the equation (4.2)

Corollary 4.3. Let f, g ∈K0 H and 0 ≤α < 1. Then f ∗g is univalent and fully starlike of order α in at least |z| < r0, where r0 = r0(α) is the real root of (4.2) in (0, 1). In particular, f ∗g is univalent and fully starlike in |z| < r0(0) ≈0.129831. Invoking Lemma 2.4 instead of Lemma 2.9 and proceeding in a similar manner to Theorem 4.2, we obtain the following result:

Theorem 4.4. Under the hypothesis of Theorem 4.2, f = h + ¯g is uni- valent and fully convex of order α in the disk |z| < s0, where s0 = s0(α) is the real root of the equation (4.3) 2(1 −α)(1 −r)5 + α(1 + r)(1 −r)2 −(1 + r)(r2 + 4r + 1) = 0 in the interval (0, 1). In particular, f is univalent and fully convex in |z| < s0(0) ≈0.0712543. It is worth remarking that the result regarding the univalence of f in

Theorem 4.4 can be further improved to 0.129831 as seen by Theorem 4.2. However, the estimate s0 given by (4.3) regarding full convexity of order α is sharp by considering the function f0(z) = 2z −(L ∗L)(z), where L is given by (1.2). In fact, as f0 has real coefficients, we obtain ∂ ∂θ  arg  ∂ ∂θf0(reiθ)  θ=0, r=s0 = 1−15s0 +15s2 0 −21s3 0 +10s4

Theorem 4.4 easily gives

Corollary 4.5. Let f, g ∈K0 H and 0 ≤α < 1. Then f ∗g is univalent and fully convex of order α in at least |z| < s0, where s0 = s0(α) is the real root of (4.3) in (0, 1). In particular, f ∗g is univalent and fully convex in |z| < s0(0) ≈0.0712543. It is expected that Corollary 4.5 can be further improved, and since the function L given by (1.2) is extremal in K0 H, in view of Example 4.1 we have the following conjecture: Conjecture C. If f, g ∈K0 H, then f ∗g is univalent and fully convex in |z|

Theorem 4.6. Let h and g have the form (1.1) with b1 = g′(0) = 0, 0 ≤α < 1 and the coefficients of the series satisfying the conditions |an| ≤1 12(n + 1)2(2n + 1) and |bn| ≤1 12(n −1)2(2n −1) for all n ≥2. Then f = h+ ¯g is univalent and fully starlike of order α in the disk |z| < r0, where r0 = r0(α) is the real root of the equation (4.4) 12(1 −α)(1 −r)5 + α(r2 + 3r + 6)(1 −r)2 −6(1 + r)3 = 0 in the interval (0, 1). In particular, f is univalent and fully starlike in |z|<r0, where r0 = r0(0) ≈0.0

Corollary 4.7. If f ∈S∗0 H and g ∈K0 H, then f ∗g is univalent and fully starlike of order α in the disk |z| < r0, where r0 = r0(α) is the real root of (4.4) in (0, 1). Acknowledgements. The research work of the first author is supported by a research fellowship from Council of Scientific and Industrial Research (CSIR), New Delhi. The authors are grateful to both the referee and the editor for their useful comments. References [1] M. Chuaqui, P. Duren and B. Osgood, Curvature properties of planar

Definition 2.1. A harmonic mapping f of the unit disk D with f(0) = 0 is said to be fully starlike of order α (0 ≤α < 1) if it maps every circle |z| = r < 1 in a one-to-one manner onto a curve that bounds a domain starlike with respect to the origin satisfying (2.1) ∂ ∂θ(arg f(reiθ)) > α, 0 ≤θ < 2π, 0 < r < 1. If α = 0, then f is fully starlike. Definition 2.2. A harmonic mapping f of the unit disk D is said to be