Abstract
The hereditary property of convexity and starlikeness for conformal mappings does not generalize to univalent harmonic mappings. This failure leads us to the notion of fully starlike and convex mappings of order α, (0\leq α<1). A bound for the radius of fully starlikeness and fully convexity of order αis determined for certain families of univalent harmonic mappings. Convexity is not preserved under the convolution of univalent harmonic convex mappings, unlike in the analytic case. Given two uni
Results & Lemmas (30)
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Theorem 2.2.
Theorem 2.2. A sense-preserving harmonic function f = h + ¯g is fully convex in D if the analytic functions h + ǫg are convex in D for each…
Theorem 2.2. A sense-preserving harmonic function f = h + ¯g is fully convex in D if the analytic functions h + ǫg are convex in D for each |ǫ| = 1.
Lemma 2.3.
Lemma 2.3. [9] Let f = h + ¯g, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n(n −α) 1 −α |an| + ∞ X n=1 n(n + α) 1 −α |bn| ≤1…
Lemma 2.3. [9] Let f = h + ¯g, where h and g are given by (1.1). Furthermore, let ∞ X n=2 n(n −α) 1 −α |an| + ∞ X n=1 n(n + α) 1 −α |bn| ≤1 and 0 ≤α < 1. Then f ∈FKH(α). The analytic description of functions in FKH(α) (0 ≤α < 1) is seen by the following
Theorem 2.4.
Theorem 2.4. Let f = h + ¯g ∈H be sense-preserving and let 0 ≤α < 1. Then f ∈FKH(α) if and only if |zh′(z)|2 Re 1 + zh′′(z) h′(z) −α…
Theorem 2.4. Let f = h + ¯g ∈H be sense-preserving and let 0 ≤α < 1. Then f ∈FKH(α) if and only if |zh′(z)|2 Re 1 + zh′′(z) h′(z) −α > |zg′(z)|2 Re
Theorem 2.7.
Theorem 2.7. A sense-preserving harmonic function f = h + ¯g is fully starlike in D if the analytic functions h + ǫg are starlike in D for…
Theorem 2.7. A sense-preserving harmonic function f = h + ¯g is fully starlike in D if the analytic functions h + ǫg are starlike in D for each |ǫ| = 1. Let FS∗ H(α) denote the subclass of S∗ H consisting of fully starlike functions of order α (0 ≤α < 1), with FS∗ H := FS∗ H(0) and let FS∗0 H (α) = FS∗ H(α) ∩S∗0 H . In [10], Jahangiri gave a sufficient condition for functions f ∈H to be in FS∗ H(α).
Lemma 2.8.
Lemma 2.8. [10] 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…
Lemma 2.8. [10] 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 2.4, the analytic characterization of functions in FS∗ H(α)
Theorem 2.9.
Theorem 2.9. 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)…
Theorem 2.9. 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
Theorem 3
Theorem 3 in [2, p. 139], f is fully starlike in D satisfying (2.3) and hence univalent by [12, Theorem 1], so f ∈FS∗ H(α). □
Theorem 3 in [2, p. 139], f is fully starlike in D satisfying (2.3) and hence univalent by [12, Theorem 1], so f ∈FS∗ H(α). □
Theorem 2.13. · radius
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…
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 an abundant examples of fully convex and fully starlike 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 are fully convex of order
Corollary 2.15.
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…
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 2.16.
Theorem 2.16. 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…
Theorem 2.16. 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.
Theorem 3.1.
Theorem 3.1. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions (1.2). Then f = h + ¯g is…
Theorem 3.1. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions (1.2). 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.
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…
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). Proceeding in a similar manner as in Theorem 3.1 and invoking Lemma 2.3 instead of
Lemma 2.8
Lemma 2.8, we have the following result.
Lemma 2.8, we have the following result.
Theorem 3.3.
Theorem 3.3. Under the hypothesis of Theorem 3.1, f = h + g is univalent and fully convex of order α in the disk |z| < rC, where rC = rC(α)…
Theorem 3.3. Under the hypothesis of Theorem 3.1, f = h + g is univalent 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 (1.3). In fact, as f0 has real coefficients, we obtain ∂ ∂θ a
Theorem 3.3
Theorem 3.3 immediately gives
Theorem 3.3 immediately gives
Corollary 3.4.
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…
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. Corresponding to
Theorem 3.1 · radius
Theorem 3.1, the next theorem determines the radius of univalence and fully starlikeness of order α for functions f = h + ¯g ∈H, where the…
Theorem 3.1, the next theorem determines the radius of univalence and fully starlikeness of order α for functions f = h + ¯g ∈H, where the Taylor coefficients of the series of h and g satisfy (1.4).
Theorem 3.5.
Theorem 3.5. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions (1.4). Then f = h + ¯g is…
Theorem 3.5. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy 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 (3.6) (2 −α)(1 −r)3 + αr(1 −r)2 −1 −r = 0 in the interval (0, 1). Moreover, this result is sharp for each α ∈[0, 1).
Corollary 3.6.
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…
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).
Theorem 2.16
Theorem 2.16 shows that the results in Corollaries 3.2 and 3.6 are not sharp if α = 0. Using Lemma 2.3 and proceeding in a similar manner…
Theorem 2.16 shows that the results in Corollaries 3.2 and 3.6 are not sharp if α = 0. Using Lemma 2.3 and proceeding in a similar manner as in Theorem 3.5, we obtain the following result.
Theorem 3.7. · radius
Theorem 3.7. Under the hypothesis of Theorem 3.5, f = h + ¯g is univalent and fully convex of order α in the disk |z| < rC, where rC =…
Theorem 3.7. Under the hypothesis of Theorem 3.5, f = h + ¯g is univalent 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 considering the function f0(z) = h0(z) + g0(z), where h0(z) = 2z −1 2 z 1 −z + z (1 −z)2
Corollary 3.8.
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…
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 (1.5) gives the sharp bound for α = 0, therefore 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 in (0,
Theorem 4.2.
Theorem 4.2. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions |an| ≤ n + 1 2 2 and |bn|…
Theorem 4.2. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions |an| ≤ n + 1 2 2 and |bn| ≤ n −1 2 2 , for all n ≥1. 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
Corollary 4.3.
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…
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 (4.1) in (0, 1). In particular f ∗g is univalent and fully starlike in |z| < r0(0) ≈0.129831. Invoking Lemma 2.3 instead of Lemma 2.8 and proceeding in a similar manner as in
Theorem 4.2
Theorem 4.2, we obtain the following result.
Theorem 4.2, we obtain the following result.
Theorem 4.4.
Theorem 4.4. Under the hypothesis of Theorem 4.2, f = h + ¯g is univalent and fully convex of order α in the disk |z| < s0, where s0 =…
Theorem 4.4. Under the hypothesis of Theorem 4.2, f = h + ¯g is univalent and fully convex of order α in the disk |z| < s0, where s0 = s0(α) is the real root of the equation (4.2) 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’s worth to remark 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 estimat
Theorem 4.4
Theorem 4.4 easily gives
Theorem 4.4 easily gives
Corollary 4.5.
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…
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 (4.2) in (0, 1). In particular f ∗g is univalent and fully convex in |z| < s0 = s0(0) ≈0.0712543. It is expected that Corollary 4.5 can be further improved, and since the function L given by (1.5) is extremal in K0 H therefore 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 c
Theorem 4.6.
Theorem 4.6. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions |an| ≤1 12(n + 1)2(2n + 1)…
Theorem 4.6. Let h and g have the form (1.1), 0 ≤α < 1 and the coefficients of the series satisfy the conditions |an| ≤1 12(n + 1)2(2n + 1) and |bn| ≤1 12(n −1)2(2n −1), for all n ≥1.
Corollary 4.7.
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…
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 (4.3) in (0, 1). References [1] D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings, Complex Anal. Oper. Theory (2011) 5:767-774, DOI 10.1007/s11785-010-0056-7. [2] M. Chuaqui, P. Duren and B. Osgood, Curvature properties of planar harmonic mappings, Comp. Metods Funct. Theory 4 (2004), no. 1, 127-142.. [3] J. Clun
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