June 15, 2026
We investigate normality and \(\varphi\)-normality criteria for harmonic mappings in the unit disk. A new sufficient condition for normality involving extended spherical derivatives is established. We further prove a Zalcman–Pang type rescaling lemma for \(\varphi\)-normal harmonic mappings and derive new \(\varphi\)-normality criteria as applications. In addition, we introduce \(\varphi\)-normal families of harmonic mappings and obtain corresponding Lappan-type characterizations. In particular, we show that for sense-preserving harmonic mappings the relevant test set may be taken to consist of only three points.
The theory of normal functions and normal families occupies a central position in geometric function theory and complex analysis. Originating in the work of Montel [1], normality has become one of the most fundamental compactness principles in the study of analytic and meromorphic functions. A family of meromorphic functions defined in a domain \(D\subseteq\mathbb{C}\) is said to be normal if every sequence in the family contains a subsequence which converges locally uniformly, with respect to the spherical metric, to a meromorphic function or to the point at infinity. One of the most important characterizations of normality is Marty’s theorem [2], which asserts that a family of meromorphic functions is normal if and only if the corresponding spherical derivatives are locally uniformly bounded. Recall that the spherical derivative of a meromorphic function \(f\) is \[\label{eq:sphmero} f^{\#}(z)=\frac{|f'(z)|}{1+|f(z)|^{2}}.\tag{1}\]
Closely related to Marty’s criterion is the notion of a normal function, introduced implicitly by Yosida [3] and Noshiro [4], and later formalized by Lehto and Virtanen [5]. A meromorphic function on the unit disk \(\mathbb{D}\) is called normal if its post-composition with conformal automorphisms of \(\mathbb{D}\) forms a normal family. This admits the following equivalent quantitative characterization.
Definition 1. A meromorphic function \(f\) in the unit disk \(\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}\) is called normal if \[\sup_{z\in\mathbb{D}}\,(1-|z|^{2})\,f^{\#}(z)<\infty.\]
In recent decades considerable efforts have been devoted to extending classical results from analytic and meromorphic function theory to harmonic mappings driven by the enormous applicability potential of harmonic mappings (see [6], [7]). Recall that a complex-valued harmonic mapping defined in a simply connected domain \(D\subseteq\mathbb{C}\) admits a canonical decomposition \(f=h+\bar g\), where \(h\) and \(g\) are analytic in \(D\) (see [8]). Throughout this paper we use the decomposition \(f=h+\bar g\).
Motivated by the work of Colonna on Bloch harmonic mappings [9], Arbeláez, Hernández and Sierra [10] introduced the notion of normality for harmonic mappings.
Definition 2. A harmonic mapping \(f=h+\bar g\) in \(\mathbb{D}\) is said to be normal in \(\mathbb{D}\) if it satisfies the Lipschitz-type condition \[\sup_{\substack{z,w\in\mathbb{D}\\ z\neq w}}\frac{\chi(f(z),f(w))}{\rho_{h}(z,w)}<\infty,\] where \(\chi(f(z),f(w))\) is the chordal distance, i.e.the Euclidean distance between the stereographic projections of \(f(z)\) and \(f(w)\) on the Riemann sphere, \[\chi(f(z),f(w))= \begin{cases} \dfrac{|f(z)-f(w)|}{\sqrt{1+|f(z)|^{2}}\,\sqrt{1+|f(w)|^{2}}}, & f(z),f(w)\in\mathbb{C},\\[2.2ex] \dfrac{1}{\sqrt{1+|f(z)|^{2}}}, & f(w)=\infty,\\[2.2ex] 0, & f(z)=f(w)=\infty, \end{cases}\] and \(\rho_{h}(z,w)\) is the hyperbolic distance between \(z\) and \(w\), \[\rho_{h}(z,w)=\frac{1}{2}\log\!\left(\frac{1+t}{1-t}\right),\qquad t=\left|\frac{z-w}{1-\bar w z}\right|.\]
The following equivalent characterization is often more convenient.
Definition 3 ([10]). A harmonic mapping \(f=h+\bar g\) in \(\mathbb{D}\) is normal in \(\mathbb{D}\) if \[\label{eq:normalharmeq} \sup_{z\in\mathbb{D}}\,(1-|z|^{2})\,\frac{|h'(z)|+|g'(z)|}{1+|f(z)|^{2}}<\infty.\tag{2}\]
The theory of normal harmonic mappings has attracted significant attention. Deng, Ponnusamy and Qiao [11] established harmonic analogues of several classical theorems on normal meromorphic functions, including versions of Marty’s theorem and the Lohwater–Pommerenke rescaling theorem. Bharti and Thin [12] established a Zalcman–Pang type rescaling result in the same setting. Following [12], the spherical derivative of a harmonic mapping \(f=h+\bar g\) in \(\mathbb{D}\) is \[f^{\#}(z)=\frac{|h'(z)|+|g'(z)|}{1+|f(z)|^{2}},\] and, for \(k\in\mathbb{N}\), the extended spherical derivative of order \(k\) is \[\label{eq:extsph} f^{\#(k)}(z):=\frac{|h^{(k)}(z)|+|g^{(k)}(z)|}{1+|f(z)|^{k+1}}.\tag{3}\]
Motivated by the generalized Marty-type theorem of Li and Xie [13] and by the normality criteria of Chen, Nevo and Pang [14] involving lower bounds of differential expressions, we first establish a new criterion for a harmonic mapping to be normal.
Theorem 1. Let \(k\) be a positive integer and let \(M>0\). Suppose that \(f=h+\bar g\) is a non-constant harmonic mapping in \(\mathbb{D}\) satisfying \[\label{eq:lowerbound} f^{\#(k)}(z)>M,\qquad z\in\mathbb{D}.\tag{4}\] Then \(f\) is a normal harmonic mapping.
Choosing \(k=1\) in Theorem 1 recovers [12]. Moreover Theorem 1 may be regarded as a harmonic analogue, for higher-order derivatives, of the result of Grahl and Nevo [15].
We next study weighted versions of normality. Our work is motivated by Aulaskari and Rättyä [16], who introduced \(\varphi\)-normal meromorphic functions. A function \(\varphi\colon[0,1)\to(0,\infty)\) is said to be smoothly increasing if \[\varphi(r)(1-r)\longrightarrow\infty\quad(r\to1^{-}),\qquad \frac{\varphi\!\left(\left|a+\dfrac{z}{\varphi(|a|)}\right|\right)}{\varphi(|a|)}\longrightarrow 1 \quad(|a|\to1^{-}),\] the latter converges uniformly on compact subsets of \(\mathbb{C}\). A typical example is \(\varphi(r)=(1-r)^{-\alpha}\), \(\alpha\in(1,\infty)\).
If \(\varphi\) is smoothly increasing then, without loss of generality, we shall assume throughout that \[\label{eq:phinorm} \varphi(r)(1-r)\ge 1,\qquad 0\le r<1.\tag{5}\] Indeed, replacing \(\varphi\) by \(\max\{\varphi,1\}\) changes neither the smoothly increasing property nor the notion of \(\varphi\)-normality below, since the two functions agree near the boundary, where the relevant estimates take place.
Given such a \(\varphi\), a meromorphic function in \(\mathbb{D}\) is \(\varphi\)-normal if \[\sup_{z\in\mathbb{D}}\frac{f^{\#}(z)}{\varphi(|z|)}<\infty.\] Following [16], Bohra, Datt and Pal [17] extended \(\varphi\)-normality to planar harmonic mappings: a harmonic mapping \(f=h+\bar g\) in \(\mathbb{D}\) is \(\varphi\)-normal if \[\sup_{z\in\mathbb{D}}\frac{f^{\#}(z)}{\varphi(|z|)}<\infty,\] where \(f^{\#}\) is as in 3 with \(k=1\).
The theory of \(\varphi\)-normal harmonic mappings is largely unexplored. Our central tool is the following Zalcman–Pang type rescaling lemma. Only the implication stated is required in the sequel; see Remark 9 for a discussion of the converse.
Theorem 2. Let \(\varphi\colon[0,1)\to(0,\infty)\) be a continuous smoothly increasing function and let \(\beta\in(-1,\infty)\). If a non-constant harmonic mapping \(f=h+\bar g\) in \(\mathbb{D}\) is not \(\varphi\)-normal, then there exist sequences \(\{w_{n}\}\subset\mathbb{D}\) and \(\{\rho_{n}\}\subset(0,1)\), with \(\rho_{n}\to0\) as \(n\to\infty\), such that the functions \[F_{n}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{\beta} f\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right)\] converge locally uniformly in \(\mathbb{C}\) to a non-constant harmonic mapping \(F\) satisfying \(F^{\#}(\eta)\le F^{\#}(0)=1\).
As a consequence we obtain the following sufficient condition for \(\varphi\)-normality.
Theorem 3. Let \(k\) be a positive integer and let \(\alpha>1\), \(M>0\). Let \(\varphi\colon[0,1)\to(0,\infty)\) be a continuous smoothly increasing function. Suppose that \(f=h+\bar g\) is a non-constant harmonic mapping in \(\mathbb{D}\) satisfying \[\label{eq:phisuff} \frac{|h^{(k)}(z)|+|g^{(k)}(z)|}{1+|f(z)|^{\alpha}}>\frac{M}{\varphi(|z|)^{\alpha-1}}, \qquad z\in\mathbb{D}.\tag{6}\] Then \(f\) is a \(\varphi\)-normal harmonic mapping in \(\mathbb{D}\).
Example 1. Let \(\varphi(r)=(1-r)^{-2}\), \(r\in[0,1)\), and let \(f(z)=z+\tfrac12\bar z\), so that \(h(z)=z\) and \(g(z)=\tfrac12 z\). Take \(\alpha=\tfrac32\) and \(M=\tfrac32\big/\big(1+(\tfrac32)^{3/2}\big)\). Since \(|h'(z)|+|g'(z)|=\tfrac32\) and \(|f(z)|\le\tfrac32|z|<\tfrac32\), we have for all \(z\in\mathbb{D}\) \[\frac{|h'(z)|+|g'(z)|}{1+|f(z)|^{3/2}} =\frac{3/2}{1+|f(z)|^{3/2}} >\frac{3/2}{1+(3/2)^{3/2}} =M\ge M(1-|z|)=\frac{M}{\varphi(|z|)^{1/2}},\] so \(f\) satisfies 6 . Moreover \[\frac{f^{\#}(z)}{\varphi(|z|)}=\frac{3/2}{1+|f(z)|^{2}}\,(1-|z|)^{2}\le\frac{3}{2},\] so \(\sup\limits_{z\in\mathbb{D}}f^{\#}(z)/\varphi(|z|)<\infty\) and \(f\) is \(\varphi\)-normal, as predicted by Theorem 3.
Corollary 1. Let \(k\) be a positive integer, \(M>0\), and let \(\varphi\colon[0,1)\to(0,\infty)\) be a continuous smoothly increasing function. If \(f=h+\bar g\) is a non-constant harmonic mapping in \(\mathbb{D}\) satisfying \[\label{eq:corphisuff} f^{\#(k)}(z)>\frac{M}{\varphi(|z|)^{k}},\qquad z\in\mathbb{D},\tag{7}\] then \(f\) is \(\varphi\)-normal.
We next pass to families. Following Lohwater and Pommerenke [18], Zalcman [19] established his rescaling lemma for normal families of meromorphic functions, since then one of the most powerful tools in the theory. We introduce the corresponding notion in the present setting.
Definition 4. A family \(\mathcal{F}\) of harmonic mappings in \(\mathbb{D}\) is said to be \(\varphi\)-normal in \(\mathbb{D}\) if for each compact set \(K\subset\mathbb{D}\) there exists \(M>0\) such that \[\sup\left\{\frac{f^{\#}(z)}{\varphi(|z|)}:z\in K,\;f\in\mathcal{F}\right\}<M.\]
Example 2. Let \(\varphi(|z|)=(1-|z|)^{-2}\) and \(\mathcal{F}=\big\{f_{n}(z)=z+\tfrac1n\,\overline{z^{\,n}}:z\in\mathbb{D}\big\}\). Then \(h(z)=z\), \(g_{n}(z)=\tfrac1n z^{n}\), so \(|h'(z)|+|g_{n}'(z)|=1+|z|^{n-1}\) and \[f_{n}^{\#}(z)=\frac{1+|z|^{\,n-1}}{1+|f_{n}(z)|^{2}}.\] Hence for every compact \(K\subset\mathbb{D}\), \[\sup_{\substack{z\in K\\ f_{n}\in\mathcal{F}}}\frac{f_{n}^{\#}(z)}{\varphi(|z|)} =\sup_{\substack{z\in K\\ f_{n}\in\mathcal{F}}} \frac{(1+|z|^{\,n-1})(1-|z|)^{2}}{1+|f_{n}(z)|^{2}}<2,\] so \(\mathcal{F}\) is a \(\varphi\)-normal family.
Theorem 4. Let \(\varphi\colon[0,1)\to(0,\infty)\) be a continuous smoothly increasing function and let \(\beta\in(-1,\infty)\). If a family \(\mathcal{F}\) of non-constant harmonic mappings in \(\mathbb{D}\) is not \(\varphi\)-normal, then there exist sequences \(\{w_{n}\}\subset\mathbb{D}\), \(\{f_{n}\}\subset\mathcal{F}\) and \(\{\rho_{n}\}\subset(0,1)\), with \(\rho_{n}\to0\), such that \[F_{n}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{\beta} f_{n}\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right)\] converge locally uniformly in \(\mathbb{C}\) to a non-constant harmonic mapping \(F\) satisfying \(F^{\#}(\eta)\le F^{\#}(0)=1\).
As an application we obtain a Lappan-type theorem. The novelty is that the test set \(E\) may be taken to have only three points (rather than five), a saving made possible by Lemma 6 below.
Theorem 5. Let \(\varphi\colon[0,1)\to(0,\infty)\) be a continuous smoothly increasing function. Let \(\mathcal{F}\) be a family of sense-preserving harmonic mappings in \(\mathbb{D}\) and let \(E\) be a set of three distinct complex numbers such that \[\sup\left\{\frac{f^{\#}(w)}{\varphi(|w|)}:w\in f^{-1}(E),\;f\in\mathcal{F}\right\}<\infty.\] Then \(\mathcal{F}\) is \(\varphi\)-normal in \(\mathbb{D}\).
Theorem 6. Let \(f=h+\bar g\) be a non-constant sense-preserving harmonic mapping in \(\mathbb{D}\) and let \(E\) be a set of three distinct complex numbers such that \[\sup\left\{\frac{f^{\#}(w)}{\varphi(|w|)}:w\in f^{-1}(E)\right\}<\infty.\] Then \(f\) is a \(\varphi\)-normal harmonic mapping in \(\mathbb{D}\).
Remark 7. Theorem 6 improves [17] by reducing the cardinality of the test set from five to three. By the same argument, the cardinality of the set \(E\) in the Lappan-type criterion [12] can likewise be reduced to three.
We collect the auxiliary results used in the proofs. The following is the harmonic analogue of Marty’s theorem, due to Deng, Ponnusamy and Qiao [11].
Lemma 1 ([11]). A family \(\mathcal{F}\) of harmonic mappings \(f=h+\bar g\) in \(\mathbb{D}\) is normal if \(\{f^{\#}(z):f\in\mathcal{F}\}\) is locally uniformly bounded.
Remark 8. A partial converse holds under the additional assumption \(\operatorname{Re}\big(h(z)\overline{g(z)}\big)>0\); see [17].
Lemma 2 ([12]). A non-constant harmonic mapping \(f\) in \(\mathbb{D}\) is normal if and only if there do not exist sequences \(\{w_{n}\}\subset\mathbb{D}\) and \(\{\rho_{n}\}\subset(0,\infty)\) with \(\rho_{n}\to0\) such that \[F_{n}(\eta):=\rho_{n}^{\,\beta}\,f(w_{n}+\rho_{n}\eta)\] converges locally uniformly in \(\mathbb{C}\) to a non-constant harmonic mapping \(F\) satisfying \(F^{\#}(\eta)\le F^{\#}(0)=1\), where \(\beta\in(-1,\infty)\) is arbitrary but fixed.
Lemma 3 ([17]). Let \(\varphi\colon[0,1)\to(0,\infty)\) be smoothly increasing. A non-constant harmonic mapping \(f=h+\bar g\) in \(\mathbb{D}\) is not \(\varphi\)-normal if and only if there exist sequences \(\{w_{n}\}\subset\mathbb{D}\) and \(\{\rho_{n}\}\subset(0,1)\), with \(\rho_{n}\to0\), such that \[F_{n}(\eta)=f\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right)\] converges locally uniformly in \(\mathbb{C}\) to a non-constant harmonic mapping \(F\).
Lemma 4 ([8]). Let \(\{f_{n}\}\) be a sequence of sense-preserving harmonic mappings in \(\mathbb{D}\) converging locally uniformly to a sense-preserving harmonic mapping \(f\). Then \(z_{0}\in\mathbb{D}\) is a zero of \(f\) if and only if \(z_{0}\) is a cluster point of the zeros of \(f_{n}\), \(n\ge1\).
The next lemma is classical (see [20]); we include a proof for completeness.
Lemma 5. Let \(f\) be a non-constant entire function. Then there are at most two values \(a\) for which all zeros of \(f-a\) are multiple.
Proof. We use the second fundamental theorem of Nevanlinna. Suppose, to the contrary, that there exist three distinct values \(a_{1},a_{2},a_{3}\in\mathbb{C}\) such that all zeros of \(f-a_{j}\) (\(j=1,2,3\)) have multiplicity at least \(2\). Since \(f\) is entire, \(\overline{N}(r,f)=0\), and since every zero of \(f-a_{j}\) is multiple, \[\overline{N}\!\big(r,\tfrac{1}{f-a_{j}}\big)\le\tfrac12 N\!\big(r,\tfrac{1}{f-a_{j}}\big) \le\tfrac12 T(r,f)+O(1).\] Hence \[2T(r,f)\le \overline{N}(r,f)+\sum_{j=1}^{3}\overline{N}\!\left(r,\frac{1}{f-a_{j}}\right)+S(r,f) \le \frac{3}{2}\,T(r,f)+S(r,f),\] so that \(\tfrac12 T(r,f)\le S(r,f)\), which is impossible. Hence the proof. ◻
The following sharpens [11], where four values appear in place of two.
Lemma 6. Let \(f=h+\bar g\) be a sense-preserving harmonic mapping in \(\mathbb{C}\) with \(g(0)=0\). Then there are at most two values \(a\) for which all zeros of \(f-a\) are multiple.
Proof. Since \(f\) is sense-preserving in \(\mathbb{C}\), we have \(|g'(z)/h'(z)|<1\) in \(\mathbb{C}\); as \(g'/h'\) is then a bounded entire function, Liouville’s theorem gives \(\omega(z)=g'(z)/h'(z)\equiv c\) with \(c\in\mathbb{C}\), \(|c|<1\). Integrating \(g'=c\,h'\) with \(g(0)=0\) yields \(g(z)=c\big(h(z)-h(0)\big)\), whence \[f(z)=h(z)+\overline{c\,h(z)-c\,h(0)}.\] As in [17], for any \(a\in\mathbb{C}\) the equation \(f(z)=a\) is equivalent to \[\label{eq:correspond} h(z)=a^{*}:=\frac{a-\overline{c\,a}+\overline{c\,h(0)}-|c|^{2}h(0)}{1-|c|^{2}}.\tag{8}\] Moreover \(f(z)-a=u(z)+\bar c\,\overline{u(z)}\) with \(u=h-a^{*}\), so \(z_{0}\) is a zero of \(f-a\) of order \(p\) if and only if it is a zero of \(h-a^{*}\) of order \(p\); in particular the multiplicities coincide. Since \(a\mapsto a^{*}\) is a bijection of \(\mathbb{C}\), and since by Lemma 5 there are at most two values \(a^{*}\) for which every zero of \(h-a^{*}\) is multiple, there are at most two values \(a\) for which every zero of \(f-a\) is multiple. ◻
Proof of Theorem 1. Suppose, to the contrary, that \(f\) is not normal. By Lemma 2 (with \(\beta=2\)), there exist sequences \(\{w_{n}\}\subset\mathbb{D}\) and \(\{\rho_{n}\}\subset(0,1)\) with \(|w_{n}|\to1^{-}\) and \(\rho_{n}\to0\) such that \[F_{n}(\eta)=\rho_{n}^{2}f(w_{n}+\rho_{n}\eta) =\rho_{n}^{2}\big(h(w_{n}+\rho_{n}\eta)+\overline{g(w_{n}+\rho_{n}\eta)}\big)\] converges locally uniformly in \(\mathbb{C}\) to a non-constant harmonic mapping \(F(\eta)=H(\eta)+\overline{G(\eta)}\) with \(F^{\#}(\eta)\le1\) and \(F^{\#}(0)=1\). Write \(H_{n}(\eta):=\rho_{n}^{2}h(w_{n}+\rho_{n}\eta)\) and \(G_{n}(\eta):=\rho_{n}^{2}g(w_{n}+\rho_{n}\eta)\). Since \(F_{n}\to F\) locally uniformly, for every positive integer \(k\) we have \(H_{n}^{(k)}\to H^{(k)}\) and \(G_{n}^{(k)}\to G^{(k)}\) locally uniformly. As \(F\) is non-constant there is \(\eta_{0}\) with \(F(\eta_{0})\neq0\).
Put \(\zeta_{n}=w_{n}+\rho_{n}\eta_{0}\). Using \(H_{n}^{(k)}(\eta_{0})=\rho_{n}^{k+2}h^{(k)}(\zeta_{n})\), \(G_{n}^{(k)}(\eta_{0})=\rho_{n}^{k+2}g^{(k)}(\zeta_{n})\) and \(|F_{n}(\eta_{0})|^{k+1}=\rho_{n}^{2(k+1)}|f(\zeta_{n})|^{k+1}\), we estimate \[\begin{align} |H_{n}^{(k)}(\eta_{0})|+|G_{n}^{(k)}(\eta_{0})| &\ge\frac{|H_{n}^{(k)}(\eta_{0})|+|G_{n}^{(k)}(\eta_{0})|}{1+|F_{n}(\eta_{0})|^{k+1}} =\frac{\rho_{n}^{k+2}\big(|h^{(k)}(\zeta_{n})|+|g^{(k)}(\zeta_{n})|\big)}{1+\rho_{n}^{2(k+1)}|f(\zeta_{n})|^{k+1}}\\[0.4ex] &=\frac{|h^{(k)}(\zeta_{n})|+|g^{(k)}(\zeta_{n})|}{\rho_{n}^{k}\,|f(\zeta_{n})|^{k+1}}\cdot \frac{|F_{n}(\eta_{0})|^{k+1}}{1+|F_{n}(\eta_{0})|^{k+1}}\\[0.4ex] &\ge\frac{|h^{(k)}(\zeta_{n})|+|g^{(k)}(\zeta_{n})|}{\rho_{n}^{k}\big(1+|f(\zeta_{n})|^{k+1}\big)}\cdot \frac{|F_{n}(\eta_{0})|^{k+1}}{1+|F_{n}(\eta_{0})|^{k+1}}\\[0.4ex] &>\frac{M}{\rho_{n}^{k}}\cdot\frac{|F_{n}(\eta_{0})|^{k+1}}{1+|F_{n}(\eta_{0})|^{k+1}}, \end{align}\] where the last inequality uses the hypothesis 4 , i.e. \(\big(|h^{(k)}|+|g^{(k)}|\big)/(1+|f|^{k+1})>M\). Since \(\rho_{n}\to0\) and \(|F_{n}(\eta_{0})|\to|F(\eta_{0})|\neq0\), the right-hand side tends to \(\infty\). But the left-hand side converges to \(|H^{(k)}(\eta_{0})|+|G^{(k)}(\eta_{0})|<\infty\), a contradiction. Hence \(f\) is normal. ◻
Proof of Theorem 2. Suppose \(f\) is not \(\varphi\)-normal. Then there is a sequence \(\{w_{n}'\}\subset\mathbb{D}\) with \(|w_{n}'|\to1^{-}\) and \[\label{eq:p1} \frac{f^{\#}(w_{n}')}{\varphi(|w_{n}'|)}\longrightarrow\infty.\tag{9}\] By the continuity of \(\varphi\) we may choose \(r_{n}\in(0,1)\) with \(|w_{n}'|<r_{n}<1\) and \[\label{eq:p2} \frac{f^{\#}(w_{n}')}{\varphi\big(|w_{n}'|/r_{n}\big)}\longrightarrow\infty.\tag{10}\] For \(t\in(0,1]\) and \(|w|<r_{n}\) define \[\Gamma_{n}(t,w):=\frac{1}{\varphi\big(|w|/r_{n}\big)^{\,1+\beta}}\cdot \frac{t^{\,1+\beta}\big(1+|f(w)|^{2}\big)f^{\#}(w)}{1+\dfrac{t^{2\beta}|f(w)|^{2}}{\varphi(|w|/r_{n})^{2\beta}}}.\] Then \(\Gamma_{n}\) is continuous on \((0,1]\times\{|w|<r_{n}\}\), and since \(\beta>-1\), \[\label{eq:p3} \lim_{t\to0}\Gamma_{n}(t,w)=0.\tag{11}\]
Claim 1. \(\displaystyle \Gamma_{n}(t,w)\ge\frac{t^{\,1+|\beta|}f^{\#}(w)}{\varphi(|w|/r_{n})^{\,1+|\beta|}}.\)
Case \(\beta\ge0\). By 5 we have \(\varphi(|w|/r_{n})\ge1\), and since \(t\le1\) this gives \(t^{2\beta}/\varphi(|w|/r_{n})^{2\beta}\le1\). Hence the denominator is at most \(1+|f(w)|^{2}\), and \[\Gamma_{n}(t,w)\ge\frac{1}{\varphi(|w|/r_{n})^{1+\beta}}\cdot \frac{t^{1+\beta}\big(1+|f(w)|^{2}\big)f^{\#}(w)}{1+|f(w)|^{2}} =\frac{t^{1+\beta}f^{\#}(w)}{\varphi(|w|/r_{n})^{1+\beta}}.\] Case \(\beta<0\). Then \(|\beta|=-\beta>0\), and the same computation with \(|\beta|\) in place of \(\beta\) yields the claim. This proves Claim 1.
By 10 and Claim 1, \[\label{eq:p4} \Gamma_{n}(1,w_{n}')\ge\frac{f^{\#}(w_{n}')}{\varphi(|w_{n}'|/r_{n})^{\,1+|\beta|}} \longrightarrow\infty.\tag{12}\] Thus \(\sup_{|w|<r_{n}}\Gamma_{n}(1,w)\ge\Gamma_{n}(1,w_{n}')>1\) for large \(n\), while by 11 \(\sup_{|w|<r_{n}}\Gamma_{n}(t,w)<1\) for \(t\) small. By continuity there exist \(t_{n}\in(0,1)\) and \(w_{n}\) with \(|w_{n}|<r_{n}\) such that \[\label{eq:p5} \sup_{|w|<r_{n}}\Gamma_{n}(t_{n},w)=\Gamma_{n}(t_{n},w_{n})=1.\tag{13}\] Combining 13 with Claim 1, \[1=\Gamma_{n}(t_{n},w_{n})\ge\Gamma_{n}(t_{n},w_{n}')\ge \frac{t_{n}^{\,1+|\beta|}f^{\#}(w_{n}')}{\varphi(|w_{n}'|/r_{n})^{\,1+|\beta|}},\] which together with 12 forces \(t_{n}\to0\).
Set \(\rho_{n}=\dfrac{t_{n}\varphi(|w_{n}|)}{\varphi(|w_{n}|/r_{n})}\). Since \(\varphi\) is increasing and \(|w_{n}|<|w_{n}|/r_{n}\), we have \(\rho_{n}<t_{n}\to0\). Define \[F_{n}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{\beta} f\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right), \qquad |\eta|<R_{n}=\frac{(1-|w_{n}|)\varphi(|w_{n}|)}{\rho_{n}}.\] A direct computation gives \[\label{eq:p7} F_{n}^{\#}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{1+\beta} \frac{\Big(1+\big|f(w_{n}+\tfrac{\rho_{n}\eta}{\varphi(|w_{n}|)})\big|^{2}\Big) f^{\#}\!\big(w_{n}+\tfrac{\rho_{n}\eta}{\varphi(|w_{n}|)}\big)}{1+\big(\tfrac{\rho_{n}}{\varphi(|w_{n}|)}\big)^{2\beta} \big|f(w_{n}+\tfrac{\rho_{n}\eta}{\varphi(|w_{n}|)})\big|^{2}},\tag{14}\] and in particular, by 13 , \[\label{eq:p8} F_{n}^{\#}(0)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{1+\beta} \frac{(1+|f(w_{n})|^{2})f^{\#}(w_{n})}{1+\big(\tfrac{\rho_{n}}{\varphi(|w_{n}|)}\big)^{2\beta}|f(w_{n})|^{2}} =\Gamma_{n}(t_{n},w_{n})=1.\tag{15}\]
Claim 2. \(\{F_{n}\}\) is normal.
Fix \(R>0\) and let \(|\eta|\le R\). By the smoothly increasing property, \[\frac{\varphi\!\Big(\big|\tfrac{w_{n}}{r_{n}}+\tfrac{\rho_{n}\eta} {\varphi(|w_{n}|/r_{n})}\big|\Big)}{\varphi(|w_{n}|/r_{n})}\longrightarrow1\] uniformly for \(|\eta|\le R\), so there is a sequence \(\delta_{n}\to0^{+}\) with, for all large \(n\), \[\label{eq:p9} \frac{(1-\delta_{n})t_{n}}{\varphi\!\big(\big|\tfrac{w_{n}}{r_{n}}+\tfrac{\rho_{n}\eta}{\varphi(|w_{n}|/r_{n})}\big|\big)} \le\frac{\rho_{n}}{\varphi(|w_{n}|)} \le\frac{(1+\delta_{n})t_{n}}{\varphi\!\big(\big|\tfrac{w_{n}}{r_{n}}+\tfrac{\rho_{n}\eta}{\varphi(|w_{n}|/r_{n})}\big|\big)}.\tag{16}\] Substituting 16 into 14 and bounding the denominator from below, \[F_{n}^{\#}(\eta)\le\frac{(1+\delta_{n})^{1+\beta}}{(1-\delta_{n})^{2\beta}} \longrightarrow1\qquad(n\to\infty),\] uniformly for \(|\eta|\le R\). Hence \(\{F_{n}^{\#}\}\) is locally uniformly bounded and, by Lemma 1, the family \(\{F_{n}\}\) is normal. This proves Claim 2.
Passing to a subsequence, \(F_{n}\to F\) locally uniformly in \(\mathbb{C}\) for some harmonic mapping \(F\). By 15 and Claim 2, \(F^{\#}(\eta)\le1\) and \(F^{\#}(0)=1\); in particular \(F^{\#}(0)\neq0\), so \(F\) is non-constant. This completes the proof. ◻
Remark 9. Only the implication proved in Theorem 2 is used below. The converse (“existence of a non-constant rescaled limit implies non-\(\varphi\)-normality”) holds for \(\beta=0\) and is precisely Lemma 3; for \(\beta\ge1\) it may fail, since the bound of the rescaled spherical derivative carries a factor of order \(\rho_{n}^{\,1-\beta}\varphi(|w_{n}|)^{\beta}\) which may tend to \(\infty.\) However, the converse holds for \(\beta\leq 0.\)
Proof of Theorem 3. Suppose, to the contrary, that \(f\) is not \(\varphi\)-normal. Applying Theorem 2 with \(\beta=\dfrac{k}{\alpha-1}+1\;(>1)\), there exist sequences \(\{w_{n}\}\subset\mathbb{D}\) and \(\{\rho_{n}\}\subset(0,1)\) with \(\rho_{n}\to0\) such that \[F_{n}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{\frac{k}{\alpha-1}+1} f\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right) =H_{n}(\eta)+\overline{G_{n}(\eta)}\] converges locally uniformly in \(\mathbb{C}\) to a non-constant harmonic mapping \(F=H+\overline{G}\) with \(F^{\#}\le1\) and \(F^{\#}(0)=1\), where \[H_{n}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{\frac{k}{\alpha-1}+1} h\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right),\quad G_{n}(\eta)=\left(\frac{\rho_{n}}{\varphi(|w_{n}|)}\right)^{\frac{k}{\alpha-1}+1} g\!\left(w_{n}+\frac{\rho_{n}\eta}{\varphi(|w_{n}|)}\right).\] Then \(H_{n}^{(k)}\to H^{(k)}\) and \(G_{n}^{(k)}\to G^{(k)}\) locally uniformly, and there is \(\eta_{0}\) with \(F(\eta_{0})\neq0\). Put \(w_{n}^{*}=w_{n}+\tfrac{\rho_{n}}{\varphi(|w_{n}|)}\eta_{0}\). Writing \(c_{n}=\rho_{n}/\varphi(|w_{n}|)\) and using \(H_{n}^{(k)}(\eta_{0})=c_{n}^{\,\frac{k}{\alpha-1}+1+k}h^{(k)}(w_{n}^{*})\) (similarly for \(G_{n}\)) and \(|F_{n}(\eta_{0})|^{\alpha}=c_{n}^{\,(\frac{k}{\alpha-1}+1)\alpha}|f(w_{n}^{*})|^{\alpha}\), \[\begin{align} |H_{n}^{(k)}(\eta_{0})|+|G_{n}^{(k)}(\eta_{0})| &\ge\frac{c_{n}^{\,\frac{k}{\alpha-1}+1+k}\big(|h^{(k)}(w_{n}^{*})|+|g^{(k)}(w_{n}^{*})|\big)}{1+|F_{n}(\eta_{0})|^{\alpha}}\\[0.4ex] &=\frac{|h^{(k)}(w_{n}^{*})|+|g^{(k)}(w_{n}^{*})|}{c_{n}^{\,\alpha-1}\,|f(w_{n}^{*})|^{\alpha}}\cdot \frac{|F_{n}(\eta_{0})|^{\alpha}}{1+|F_{n}(\eta_{0})|^{\alpha}}\\[0.4ex] &\ge\frac{|h^{(k)}(w_{n}^{*})|+|g^{(k)}(w_{n}^{*})|}{c_{n}^{\,\alpha-1}\big(1+|f(w_{n}^{*})|^{\alpha}\big)}\cdot \frac{|F_{n}(\eta_{0})|^{\alpha}}{1+|F_{n}(\eta_{0})|^{\alpha}}\\[0.4ex] &>\frac{M}{\varphi(|w_{n}^{*}|)^{\alpha-1}\,c_{n}^{\,\alpha-1}}\cdot \frac{|F_{n}(\eta_{0})|^{\alpha}}{1+|F_{n}(\eta_{0})|^{\alpha}}\\[0.4ex] &=\frac{M}{\big(\varphi(|w_{n}^{*}|)/\varphi(|w_{n}|)\big)^{\alpha-1}\rho_{n}^{\,\alpha-1}}\cdot \frac{|F_{n}(\eta_{0})|^{\alpha}}{1+|F_{n}(\eta_{0})|^{\alpha}}, \end{align}\] where we used the hypothesis 6 at \(z=w_{n}^{*}\) and \(c_{n}^{\,\alpha-1}\varphi(|w_{n}^{*}|)^{\alpha-1} =\rho_{n}^{\,\alpha-1}\big(\varphi(|w_{n}^{*}|)/\varphi(|w_{n}|)\big)^{\alpha-1}\). Since \(\varphi(|w_{n}^{*}|)/\varphi(|w_{n}|)\to1\), \(\rho_{n}\to0\) and \(|F_{n}(\eta_{0})|\to|F(\eta_{0})|\neq0\), the right-hand side tends to \(\infty\), while the left-hand side tends to \(|H^{(k)}(\eta_{0})|+|G^{(k)}(\eta_{0})|<\infty\), a contradiction. Hence \(f\) is \(\varphi\)-normal. ◻
Proof of Corollary 1. Apply Theorem 3 with \(\alpha=k+1\): then \(\alpha-1=k\) and 7 is exactly 6 . ◻
Proof of Theorem 4. The argument is identical to that of Theorem 2, with \(f\) replaced by the members \(f_{n}\) of \(\mathcal{F}\). Concretely, since \(\mathcal{F}\) is not \(\varphi\)-normal there exist \(r'\in(0,1)\), \(\{w_{n}'\}\subset\{|w|\le r'\}\) and \(\{f_{n}\}\subset\mathcal{F}\) with \(f_{n}^{\#}(w_{n}')/\varphi(|w_{n}'|)\to\infty\); choosing \(r_{n}\in(0,1)\) with \(|w_{n}'|<r_{n}<r'\) and replacing \(f^{\#}\), \(f\) by \(f_{n}^{\#}\), \(f_{n}\) throughout the definition of \(\Gamma_{n}\), Claims 1 and 2 and the construction of \(w_{n},\rho_{n}\) go through verbatim, producing the asserted limit \(F\) with \(F^{\#}\le F^{\#}(0)=1\). ◻
Proof of Theorem 5. Suppose \(\mathcal{F}\) is not \(\varphi\)-normal in \(\mathbb{D}\). By Theorem 4 (with \(\beta=0\)) there exist \(\{w_{n}\}\), \(\{\rho_{n}\}\subset(0,1)\) with \(\rho_{n}\to0\) and \(\{f_{n}\}\subset\mathcal{F}\) such that \[F_{n}(\eta)=f_{n}\!\left(w_{n}+\frac{\rho_{n}}{\varphi(|w_{n}|)}\eta\right)\longrightarrow F(\eta)\] locally uniformly in \(\mathbb{C}\), where \(F\) is a non-constant sense-preserving harmonic mapping; since \(g(0)=0\) is a value-independent normalization, Lemma 6 applies to \(F\).
Let \(b\in\mathbb{C}\) be such that \(F-b\) has a simple zero at some \(\eta_{0}\), i.e. \(F^{\#}(\eta_{0})\neq0\). By Lemma 4 there exist \(\eta_{n}\to\eta_{0}\) with \(F_{n}(\eta_{n})=f_{n}(w_{n}^{*})=b\) for all large \(n\), where \(w_{n}^{*}=w_{n}+\tfrac{\rho_{n}}{\varphi(|w_{n}|)}\eta_{n}\). Since \(F_{n}\to F\) locally uniformly, \[F_{n}^{\#}(\eta_{n})=\frac{\rho_{n}}{\varphi(|w_{n}|)}\,f_{n}^{\#}(w_{n}^{*}) \longrightarrow F^{\#}(\eta_{0})\neq0,\] and therefore, using \(\varphi(|w_{n}^{*}|)/\varphi(|w_{n}|)\to1\) and \(\rho_{n}\to0\), \[\frac{f_{n}^{\#}(w_{n}^{*})}{\varphi(|w_{n}^{*}|)} =\frac{\varphi(|w_{n}|)}{\rho_{n}}\cdot\frac{F_{n}^{\#}(\eta_{n})}{\varphi(|w_{n}^{*}|)} \longrightarrow\infty.\] Thus, whenever \(b\) is a simple value of \(F\), \[\label{eq:lappanblow} \sup_{w\in f_{n}^{-1}(\{b\})}\frac{f_{n}^{\#}(w)}{\varphi(|w|)}=\infty.\tag{17}\] By Lemma 6 there are at most two values \(b\) for which every zero of \(F-b\) is multiple. Hence, among the three distinct numbers of \(E\), at least one, say \(b_{0}\), is a simple value of \(F\), and 17 for \(b_{0}\) contradicts the hypothesis \[\sup\left\{\frac{f^{\#}(w)}{\varphi(|w|)}:w\in f^{-1}(E),\;f\in\mathcal{F}\right\}<\infty.\] Therefore \(\mathcal{F}\) is \(\varphi\)-normal in \(\mathbb{D}\). ◻
Proof of Theorem 6. The proof is the single-mapping specialization of that of Theorem 5, using Lemma 3 (the \(\beta=0\) rescaling for a single mapping) in place of Theorem 4. ◻
The second author gratefully acknowledges the Department of Science and Technology (DST), Government of India, for financial support through the DST-INSPIRE Fellowship
(No.DST/INSPIRE/03/2022/005759: IF 220689).
Conflict of interest. The authors declare that there are no conflicts of interest regarding the publication of this paper.
Data availability. Data sharing is not applicable, as the article is purely theoretical.
Kuldeep Singh Charak, Department of Mathematics, University of Jammu, Jammu-180006, India.
Email: kscharak7@rediffmail.com
Pratiksha, Department of Mathematics, University of Jammu, Jammu-180006, India.
Email: pratikshapakhetra1999@gmail.com
Nikhil Bharti, Higher Education Department, Government of Jammu and Kashmir, India;
Department of Mathematics, Government Degree College Basohli, Basohli-184201, Kathua, Jammu and Kashmir, India.
Email: nikhilbharti94@gmail.com
\(^*\) Corresponding author: Nikhil Bharti↩︎