On the pre-Schwarzian norm estimate of special close-to-convex harmonic mappings

Abstract: In this article, we consider a class of close-to-convex harmonic mappings with special analytic part in the unit disk $\mathbb{D}={z\in\mathbb{C}:|z|<1}$ and obtain sharp pre-Schwarzian norm estimate. We study the theory of harmonic Bloch mapping for the considered class. Moreover, we discuss some growth and distortion theorems for analytic and co-analytic parts of such harmonic mappings.

• The paper establishes the sharp estimate ||P_f|| ≤ 4(1−c)+1 for −1/2 ≤ c ≤ 0, with the bound decreasing from 5 to 3 as c varies across the class.
• It shows that finite pre-Schwarzian norm does not guarantee the harmonic Bloch property, revealing distinct boundary-growth behavior for these mappings.
• It derives sharp analytic and co-analytic derivative and growth bounds, including hypergeometric-function formulas that quantify how the dilatation controls distortion.

This paper by Sushil Pandit establishes a sharp pre-Schwarzian norm estimate for a one-parameter family of close-to-convex harmonic mappings on the unit disk $\mathbb{D}$, and complements this estimate with a study of the Bloch property and of growth and distortion bounds for the analytic and co-analytic parts (2606.19022).

The class under consideration

The paper works with sense-preserving harmonic mappings $f = h + \overline{g}$ normalized as $h(z) = z + \sum_{n\ge 2} a_n z^n$, $g(z) = \sum_{n\ge 1} b_n z^n$, whose dilatation is $\omega = g'/h'$. The analytic part is required to lie in

$$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$

and the resulting harmonic family is denoted $\mathcal{H}_c$. This class interpolates between known settings: $c = 0$ recovers mappings with convex analytic part, while $c = -1/2$ corresponds to the class studied by Bshouty and Lyzzaik, obtained from Umezawa's convex-in-one-direction family $\mathcal{F}$ in the limit $f = h + \overline{g}$0. Univalence and close-to-convexity of members of $f = h + \overline{g}$1 follow from earlier criteria of Bshouty–Joshi–Joshi (under $f = h + \overline{g}$2) and Ponnusamy–Kaliraj (under a condition on $f = h + \overline{g}$3), so the paper operates entirely within the univalent close-to-convex regime.

Sharp pre-Schwarzian norm estimate

The central result uses the Hernández–Martín pre-Schwarzian derivative for locally univalent harmonic mappings,

$f = h + \overline{g}$4

with norm $f = h + \overline{g}$5.

Theorem 2.1: For $f = h + \overline{g}$6, $f = h + \overline{g}$7,

$f = h + \overline{g}$8

and the bound is sharp.

The proof proceeds in two steps. First, subordination gives the sharp pointwise bound $f = h + \overline{g}$9, attained by the extremal function $h(z) = z + \sum_{n\ge 2} a_n z^n$0. Combining this with the Schwarz–Pick lemma applied to the dilatation term yields the upper bound. Second, sharpness is demonstrated via the mappings $h(z) = z + \sum_{n\ge 2} a_n z^n$1 built from the same extremal analytic part with automorphic dilatation $h(z) = z + \sum_{n\ge 2} a_n z^n$2, $h(z) = z + \sum_{n\ge 2} a_n z^n$3. Restricting to the positive real axis produces a one-variable expression $h(z) = z + \sum_{n\ge 2} a_n z^n$4 whose stationary-point analysis shows that $h(z) = z + \sum_{n\ge 2} a_n z^n$5 as $h(z) = z + \sum_{n\ge 2} a_n z^n$6.

Two consequences deserve emphasis:

• Recovery of the known constant: setting $h(z) = z + \sum_{n\ge 2} a_n z^n$7 yields $h(z) = z + \sum_{n\ge 2} a_n z^n$8 for mappings with convex analytic part, reproducing the sharp Hernández–Martín estimate. The theorem therefore extends a previously isolated result to a full continuum of classes.
• Continuum of constants: the bound ranges continuously from $h(z) = z + \sum_{n\ge 2} a_n z^n$9 (at $g(z) = \sum_{n\ge 1} b_n z^n$0) down to $g(z) = \sum_{n\ge 1} b_n z^n$1 (at $g(z) = \sum_{n\ge 1} b_n z^n$2). This contrasts with the situation for stable harmonic mappings studied by Liu and Ponnusamy, where the connection between $g(z) = \sum_{n\ge 1} b_n z^n$3 and the norm of the analytic part was also exploited; here the extremal construction via a moving automorphic dilatation is the distinguishing technique, following the approach introduced by Ali and Pandit.

Failure of the Bloch property

A notable negative result: a mapping $g(z) = \sum_{n\ge 1} b_n z^n$4 need not be a harmonic Bloch mapping. Taking the extremal analytic part $g(z) = \sum_{n\ge 1} b_n z^n$5 with dilatation $g(z) = \sum_{n\ge 1} b_n z^n$6, the quantity $g(z) = \sum_{n\ge 1} b_n z^n$7 behaves like $g(z) = \sum_{n\ge 1} b_n z^n$8 along the real axis, which is unbounded since $g(z) = \sum_{n\ge 1} b_n z^n$9 for $\omega = g'/h'$0. Thus membership in $\omega = g'/h'$1 imposes no Bloch-type control on the derivative growth near the boundary, even though the pre-Schwarzian norm remains finite. This separation between finite pre-Schwarzian norm and the Bloch property is consistent with the classical analytic picture but is made explicit here for the first time for this class.

Growth and distortion estimates

Derivative estimates: for $\omega = g'/h'$2,

$\omega = g'/h'$3

The lower and upper bounds for $\omega = g'/h'$4 follow from Suffridge's subordination theorem applied to $\omega = g'/h'$5, and are sharp at $\omega = g'/h'$6 and $\omega = g'/h'$7 respectively for the standard extremal function. For the co-analytic part, the Schwarz–Pick estimate $\omega = g'/h'$8 with $\omega = g'/h'$9 is combined with the bound on $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$0; the lower bound degenerates to $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$1 at $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$2 and the upper bound is attained as $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$3. Both extremes are realized by the extremal pair with dilatation $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$4.

Co-analytic growth: integrating $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$5 along radial paths yields sharp bounds on $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$6 expressed through the Gauss hypergeometric function $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$7. For $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$8 the bound involves $$\mathcal{A}_c = \left\{\phi \in \mathcal{A}: \operatorname{Re}\left(1 + \frac{z\phi''(z)}{\phi'(z)}\right) > c\right\}, \qquad -1/2 \le c \le 0,$$9 with $\mathcal{H}_c$0, while for $\mathcal{H}_c$1 it simplifies to a closed form in powers of $\mathcal{H}_c$2. Sharpness again holds for the canonical extremal mapping. These are distortion-type results for the co-analytic part, which are generally harder to obtain than their analytic counterparts because the dilatation enters the integration.

Limitations and open questions

Several restrictions should be noted. The definition of $\mathcal{H}_c$3 used throughout requires only local univalence and nonvanishing Jacobian, so no restriction on the dilatation beyond $\mathcal{H}_c$4 is needed — an advantage over the Kanas–Klimek-Śmęt formulation, which requires $\mathcal{H}_c$5. However, the sharpness argument relies on dilatations that are disk automorphisms approaching the boundary; whether the supremum $\mathcal{H}_c$6 is actually attained by any fixed member of $\mathcal{H}_c$7 (rather than approached in the limit) is not addressed. The co-analytic growth theorem assumes a specific structure on $\mathcal{H}_c$8 only through $\mathcal{H}_c$9, which is optimal by Schwarz–Pick, but the corresponding coefficient bounds for $c = 0$0 are not derived. Finally, the paper does not treat the Schwarzian derivative analogue or the case $c = 0$1, where the close-to-convexity criteria cited above may fail; extending the estimate outside $c = 0$2 remains open.

Conclusion

The paper delivers a sharp, parameter-dependent pre-Schwarzian norm estimate $c = 0$3 for close-to-convex harmonic mappings with analytic part satisfying $c = 0$4, unifying and generalizing the known constant $c = 0$5 for convex-analytic-part mappings. It further shows that finiteness of this norm does not imply the harmonic Bloch property within the class, and provides sharp growth and distortion estimates for both the analytic and co-analytic parts, the latter expressed via Gauss hypergeometric functions. The results sharpen the picture of how boundary behavior of the analytic part controls the geometry of the associated harmonic mappings.