Papers

Let \( \Lambda=\{\lambda_j\}_{j\in\mathbb N}\subset[0,1) \) be a finite union of Carleson sequences, and let \( \Lambda^p=\{\lambda_j^p\}_{j\in\mathbb N} \) for \(p>0\). We prove that \(\Lambda\) and \(\Lambda^p\) admit compatible Hartmann-type cluster decompositions and that, under the natural coordinate identification, \( H^2(\Lambda)=H^2(\Lambda^p). \) After taking account of the change in the normalization weights, this trace-space identity gives \( \operatorname{Ran}(T_\Lambda) = \operatorname{Ran}(T_{\Lambda^p}) \) for the associated weighted evaluation operators. The characterization of finitely generated dynamical frames for normal operators then implies that, for every positive operator \(A\in\mathcal B(H)\), every finite index set \(I\), and every \(p>0\), \( \{A^nf_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H\) if and only if \( \{A^{pn}f_i\}_{n\in\mathbb N,\,i\in I} \text{ is a frame for }H. \) In particular, for every \(q\in\mathbb N^+\), the subfamily \( \{A^{qn}f_i\}_{n\in\mathbb N,\,i\in I} \) is still a frame, and hence every such dynamical frame has infinite excess. An example shows that the positivity assumption cannot in general be replaced by normality.

We prove that the Fermat equation $f^n+g^n=h^n$ with exponent $n\geq 4$ has no nontrivial solutions $f,g,h$ in the star-invariant subspace $K_\theta^p$, in which $1\leq p\leq\infty$, whenever the boundary spectrum of the inner function $\theta$ has logarithmic capacity zero. This gives a positive answer, for this class of inner functions and exponents, to a problem posed by Dyakonov.

Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line, but scalar operations suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics using script chain complexes. The cell-count polynomial satisfies $$ \mathcal M_S(t)=\mathcal P_{S,\mathbb K}(t)+(1+t)\mathcal B_{S,\mathbb K}(t), $$ where $\mathcal P$ is the Poincar'e polynomial and $\mathcal B$ records boundary ranks. Over $\mathbb Z$, Smith labels separate unit pairs, representing refinement overhead, from non-unit pairs carrying torsion. We distinguish Cartesian products from bundles, derive the monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction. Mapping cones, joins, and double mapping cylinders arise as reduced models. We apply the construction to bivector discrepancies in Sommen's calculations. In dimension four, Hodge decomposition identifies the unit bivector sphere with $S^2*S^2$; the angular excess is a contractible relative complex from inserting the rank-two midpoint. In dimension five, the $SO(5)$-action on $S(\Lambda^2\mathbb R^5)=S^9$ has principal orbit $SO(5)/T^2$ and singular orbits $\widetilde G_2(\mathbb R^5)$ and $\mathbb{CP}^3$. For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from sphere homology by seven unit-labelled pairs. We also treat Borel–Moore realizations of selected noncompact symbols, define division as an action-certified partial operation, and describe the orbit-type face diagram for unit bivectors in dimension six. There the correction polynomial has nonnegative coefficients, while compatibility of the face attachments remains open. These examples separate subdivision and attachment data from quotient actions and support conditions suppressed by scalar notation.

We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of Fernández~de~Bobadilla and Kollár, a question of Kollár and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes. Version 2 is a revised version of sections 1–6 of the original, no change in the results. A substantially expanded version of sections 7–8 of the original will be posted separately.

This paper establishes Turán-type inequalities for quaternionic canonical generalized polynomials with all zeros in the closed unit ball of radius $k \geq 1$. We extend the classical inequality due to Govil from the complex setting to the quaternionic framework. We prove that for certain classes of polynomials, the inequality \[\|P'\| \geq \frac{n}{1+k^n}\|P\|\] holds for all $k \geq 1$, where $n$ is the degree of the polynomial. Our main results provide sharp derivative estimates for quaternionic canonical generalized polynomials under specific algebraic conditions on their zeros. These findings contribute to the ongoing research of extending classical polynomial inequalities to the noncommutative setting of quaternions.

We prove a finite-point version of the Schwarz-Pick inequality for holomorphic self-maps of the disk. The estimate compares Bergman Gram matrices built from several points and their images, and the sharp constant depends on both the number of points and, for finite Blaschke products, the degree of the map. These matrices also arise from the Szegő-kernel metric on the symmetrized polydisc, where the estimate becomes a sharp Schwarz-Pick inequality for the induced holomorphic maps.

We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\mathbb{C}$. Let $\mu=\nu+\eta$, where $\nu$ is a strongly tempered measure on $\mathbb{R}$, $\eta=\sum_{m\geq1} b(\gamma_m)\delta_{\gamma_m}$ is a strongly tempered pure point measure supported off the real line, and $a:\Lambda\rightarrow\mathbb{C}$, with $\Lambda=\{\lambda_n\}_{n\geq1}\subset\mathbb{R}$, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form \begin{align} \sum_{n\geq1} a(\lambda_n)\varphi(\lambda_n)=\int_{\mathbb{R}} \widehat{\varphi}(t)\mathrm{d}\nu(t)+\sum_{m\geq1} b(\gamma_m)\widehat{\varphi}(\gamma_m), \end{align} valid for every $\varphi\in C^\infty_c(\mathbb{R})$, where $\widehat{\varphi}$ denotes the Fourier transform. We prove that each such identity determines a unique generating function $F$ that is holomorphic and almost periodic in the half-plane above the strip and admits a meromorphic continuation to $\mathbb{C}^+$. The measure $\eta$ encodes the poles and residues of $F$, while $\nu$ describes the boundary behavior of its regular part through a generalized Nevanlinna representation, and $a$ determines its Fourier coefficients. Conversely, every function in the corresponding meromorphic class whose Fourier coefficients satisfy a local summability condition determines a unique summation identity of this form. The proof combines a strip version of the Bridge Lemma with a Cauchy-transform argument that accounts for the off-real poles. As an application, we show that the Guinand-Weil explicit formula for every member of the Selberg class, including non-self-dual members, fits into our framework, and we identify its associated generating function.

We study $L^\infty$ Widom factors for Chebyshev and residual polynomials on compact non-polar subsets of the real line that need not be regular in the sense of potential theory and, in particular, on sets with isolated points. We first show that boundedness of the Widom factors is independent of the normalization point $x_*\in\overline{\mathbb{R}}\backslash\mathsf{E}$. For semi-regular sets, meaning that the set of regular points $\mathsf{E}^\mathrm{reg}$ is closed, we introduce the irregularity coefficient $\mathcal{IR}(\mathsf{E},x_*)=\sum_{x\in\mathsf{E}\backslash\mathsf{E}^\mathrm{reg}}G_\mathsf{E}(x,x_*)$, which measures the contribution of irregular boundary points to the extremal polynomial problem. Our upper bound is \[ \sup_{n\ge1}\mathcal{W}_n(\mathsf{E},x_*) \le 2\exp\bigl[\mathcal{PW}(\mathsf{E}^\mathrm{reg},x_*)+\mathcal{IR}(\mathsf{E},x_*)\bigr], \] while the complementary lower bound is \[ \liminf_{n\to\infty}\mathcal{W}_n(\mathsf{E},x_*) \ge 2\exp\bigl[\mathcal{IR}(\mathsf{E},x_*)\bigr]. \] Consequently, for semi-regular Parreau–Widom sets the Widom factors are bounded if and only if $\mathcal{IR}(\mathsf{E},x_*)<\infty$. When the regular part is a finite union of intervals, this condition is equivalent to a geometric square-root summability condition on the irregular points. In the special case $\mathsf{E}=[a,b]\cup\{x_k\}_{k\ge1}$, the Widom factors have the exact, possibly infinite, limit $2\exp[\mathcal{IR}(\mathsf{E},x_*)]$. When this limit is finite, we obtain the corresponding Szegő–Widom asymptotics for the normalized extremal polynomials.

We investigate Bohr-type inequalities for bounded analytic functions, particularly in the presence of a parameter or under convex combination structures. Most of our findings are sharp, and we also generalize a number of earlier results.

For every $n\geq1$ and $\gamma>-1$, we construct $f\in L^1(\mathbb B_n,dv_\gamma)$ whose Toeplitz form extends to a bounded, noncompact operator on the weighted Bergman spaces $A^2_\gamma(\mathbb B_n)$ and whose Berezin transform vanishes at the boundary. Boundary vanishing of the Berezin transform therefore does not imply compactness for Toeplitz operators with integrable symbols, in contrast with operators in the Toeplitz algebra generated by bounded symbols; this answers a question of Bauer and Isralowitz. The construction approximates rank-one operators in norm by Toeplitz operators with smooth, compactly supported symbols and transports suitably separated blocks toward the boundary. On the unweighted disk, a diagonal version produces a real symbol with the same vanishing and noncompactness properties, for which the operator is positive and Zorboska's two-sided localization condition holds at $p=3$. Her hypothesis $p>3$ is shown to be sharp.

We give a complete characterization of the nef, effective, pseudo-effective, and semi-positive cones on Hopf manifolds. In particular, by computing the Ueda classes on non-diagonal Hopf surfaces, we show that the invariant elliptic curve is nef but not semi-positive. Via Bott–Chern cohomology calculations, we construct new examples of nef but non-semi-positive line bundles on Hopf manifolds of arbitrary dimension. We also prove that a minimal compact complex surface \(X\) containing an elliptic curve \(C\) and a nef line bundle \(L\) such that \((X,C,L)\) has finite generalized Ueda type is either a Hopf surface or Serre's example.

Building on prior results [1], we introduce vector-valued modular resurgent series, whose components exhibit a single infinite tower of singularities in the Borel plane, trivial secondary resurgent series, and Stokes constants given by linear combinations of the coefficients of a vector of $L$-functions. We extend the paradigm of modular resurgence to this setting, emphasizing the role of the Stokes constants and the interplay between the associated vectors of $q$-series and Dirichlet series, and describing the resulting symmetry relating canonical pairs of vector-valued modular resurgent series. Moreover, we conjecture that certain vectors of $q$-series with modular resurgent asymptotics are vector-valued quantum modular forms and can be reconstructed via median resummation. Finally, we show that vectors of $q$-Pochhammer symbols, previously considered in [2], and Eichler integrals of vector-valued modular forms of weight $1/2$ and $3/2$ can be studied within the framework of vector-valued modular resurgence. While the first case amounts to a convenient repackaging of scalar modular resurgence, the second involves a non-trivial representation of the modular group and therefore illustrates the necessity of the vector-valued framework; we establish its modular resurgent structure in general, and work it out in full detail for the unary theta series, whose Eichler integrals are the false theta functions of quantum topology.

We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram–harmonic normal form by Howe separation. A label–Capelli inverse normalizes the Clifford contractions, while a finite Wick–Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford–Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin–Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.

We settle two endpoint problems for Beurling–Carleson support conditions. For $0<s<\frac12$ and $\theta=\frac{1-2s}{1-s}$, we construct a nonatomic singular probability measure $\mu$, supported on a single $\theta$-Beurling–Carleson set, such that $S_\nu'\notin H^s$ for every nonzero submeasure $0<\nu\leq\mu$. For $m>3$ and $\alpha=\frac{m-3}{m-1}$, we construct a nonatomic probability measure supported on a single $\alpha$-Beurling–Carleson set that is not the deficiency measure of any nearly maximal solution of $\Delta u=(u_+)^m$. Thus hereditary failure persists at the first endpoint, while the critical support condition in the second problem is necessary but not sufficient. The proofs combine endpoint Cantor–Moran constructions with kernel divergence and a nonlinear energy obstruction.

We study interlacing between real polynomial quotients obtained from centered binomial samples of a fixed even or odd real entire function at consecutive odd degrees. For sources of order at most one, we determine the optimal vertical-strip half-widths guaranteeing weak interlacing for every adjacent pair: $\sqrt{15/28}$ for even sources and the explicit algebraic constant $h_*\approx 1.0656$ for odd sources. These constants are forced by the first nontrivial pair and suffice at every degree. Under the same sharp bounds, nonpolynomial sources have simple zeros in $(-2,2)$ and strict interlacing at every positive index. We also prove an eventual result: if a nonpolynomial source has growth order less than two and only finitely many zeros off the imaginary axis, then all sufficiently high adjacent quotients have simple interior zeros and strictly interlace; the starting index depends only on the finite factor formed from those zeros. The proof turns the sampling problem into an interval real-rootedness problem for a Jacobi differential operator. For fixed degree, we also determine the maximal range of scalar shifts preserving real-rooted polynomials with at most one zero outside the interval. Applications give strict interlacing for quotients attached to completed Dedekind zeta derivatives and to successive centered blocks of critical values of self-dual newforms satisfying explicit spacing and parity conditions.

Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$, and let \[ \mathcal E(h)_x=\{F\in\mathcal O(E)_x:|F|_h^2\in L^1_{\mathrm{loc},x}\} \] be its multiplier submodule sheaf. We introduce a finite plurisubharmonic Gram scalarization condition under which the higher-rank integrability problem reduces to finitely many scalar multiplier ideals. This reduction yields coherence and strong openness without imposing a general positivity hypothesis. When the scalar weights and the varying weight have analytic singularities, it also gives a theory of module jumping numbers, including a simultaneous-residue criterion for actual jumps. Finally, we study the induced Skoda filtration: Artin–Rees yields eventual periodicity, Tor controls whether periodicity starts at the scalar threshold, and a direct-image quotient measures the obstruction to descent.

We prove that every map $f:U\to\GG$ preserving the Poincaré distance and the Kobayashi distance is holomorphic or anti-holomorphic, where $U$ is a connected open subset of the unit disc. We also study the nonroyal automorphism orbits of $\GG$. A $C^1$ map on a connected relatively open part of such an orbit which preserves the restriction of the ambient Kobayashi–Royden metric is the restriction of a global automorphism or anti-automorphism of $\GG$. The same conclusion holds for maps preserving the ambient Kobayashi distance.

Chavan and Zwonek recently proved that every $C^1$ Kobayashi distance isometry of the diamond is holomorphic or antiholomorphic. We show that the $C^1$ assumption is superfluous.

This paper reviews the subject of equivalence after extension in the context of Toeplitz operators, truncated Toeplitz operators, and further generalizations including dual and multiband truncated Toeplitz operators. Paired operators are shown to play a key role here. Properties under investigation by these methods include the invertibility and Fredholm properties of generalized Toeplitz operators, with new results showing how the kernels of such operators can often be fully described given only limited information.

We give explicit examples of holomorphic submersions $p\colon Y\to X$ such that $Y$ and $X$ are complex manifolds, all the fibers of $p$ are biholomorphic to the unit disc in the complex plane, but the mapping $p$ is not holomorphically locally trivial. In our examples the complex manifolds $Y$ are neighborhoods of the zero section of a line bundle $L$ on $X$; the line bundle $L$ is not, in general, supposed to be positive or negative.

In this paper, we provide a sufficient condition under which the one-dimensional local zeta function admits a holomorphic continuation to the complex plane cut along the negative real axis.

We study Bloch and Landau type theorems for a class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying the inhomogeneous analytic dilatation equation $$ g'(z)=\omega(z)h'(z)+\psi(z),$$ where $\omega$ and $\psi$ are analytic functions in $\mathbb{D}$ with $\|\omega\|_{\infty}\leq k<1$ and $\|\psi\|_{\infty}\leq M$. Here $h$ and $g$ are called analytic and co-analytic part of $f$, respectively. We first establish a Bloch type theorem for certain normalized class of harmonic functions under the condition $k+M<1$. Finally, we obtain two versions of the Landau theorem under additional assumptions: one for bounded harmonic mappings and another under the assumption that the analytic part of a harmonic functions has bounded Bloch seminorm.

In this paper, we construct the octonionic Hessian complex, which is the octonionic version of the $\partial\bar \partial$-complex. By proving its ellipticity, we can solve the non-homogeneous octonionic Hessian equations under a compatibility condition. This is the octonionic version of the $\partial\bar\partial$-lemma. We also obtain the boundary version of octonionic Hessian operator and introduce the notion of a tangentially octonionic pluriharmonic function, which allows us to establish the Hartogs–Bochner extension for tangentially octonionic pluriharmonic functions on the boundary of a domain with connected complement. This solves the octonionic version of Poincaré's problem for pluriharmonic extension on complex space. The main difficulty of this construction is the lack of associativity in octonionic case. However, we can overcome it by careful use of weak form of associativity including Moufang identities and alternativity.

We obtain sharp bounds for the third- and fourth-order Schippers functionals, $|\sigma_3(f)(0)|$ and $|\sigma_4(f)(0)|$, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class $\mathcal{S}_{\leftmoon}^*$ and the lune-convex class $\mathcal{C}_{\leftmoon}$ determined by the subordination \[ \frac{zf'(z)}{f(z)} \prec z+\sqrt{1+z^2}, \qquad 1+\frac{zf''(z)}{f'(z)} \prec z+\sqrt{1+z^2}, \] respectively, together with the bean-domain class $\mathcal{BT}_{\mathfrak{B}}$ associated with \[ \mathfrak{B}(z)=\sqrt{1+\tanh z}. \] Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients $g_{1,1}$ and $g_{1,2}$. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.

This is a sequel to \cite{PaperI}, where it is shown that a discrete effective divisor on the open unit disk $\D$ is the zero divisor of a holomorphic function with integer Taylor coefficients if and only if it is invariant under complex conjugation. Building on that realization theorem and on the unit characterization established there, we study the prime and maximal spectra of the rings $R = \mathbb{Z}[i][[z]] \cap O(\mathbb{D})$ and $R_{\mathbb{R}} = \mathbb{Z}[[z]] \cap O(\mathbb{D})$. We give a trichotomy for maximal ideals, classify the prime ideals lying outside the class $\mathfrak{P}_1$ of primes containing a unit-constant-term element, attach to each element of $\mathfrak{P}_1$ a complete analytic invariant, and construct an injection from ultrafilters on admissible divisors into $\mathfrak{P}_1$ by an ultraproduct device. We prove the point-evaluation ideals $P_a$ are prime for all $a \in \D$ and maximal for real $a$, and deduce a conditional Bézout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.

We study holomorphic functions on permutation-invariant subsets of an infinite-dimensional sequence space that are invariant under all finite permutations of coordinates. Working on domains modeled on the space $\czero$ of null sequences, we show that permutation symmetry forces the first differential to vanish along every constant tail: the coordinate derivatives agree there and an averaging argument annihilates their common value. This yields rigidity—constancy—on domains of constant sequences, and more generally on any $\cc$-chain-connected domain on which the differential vanishes on the finitely supported directions. The method has a sharp boundary: a precise obstruction prevents the elementary argument from reaching the finitely many exceptional coordinates of a non-constant point, so unconditional rigidity on general eventually-constant domains remains open. In the other direction, an explicit example—any Banach limit on $\ell^\infty$—shows that the chain-connectedness and boundedness hypotheses cannot be dropped: mere topological connectedness admits non-constant symmetric entire functions. We record the vector-valued corollary and discuss the motivating application to holomorphic cross-sections of the zero-set map, isolating the lifting problem that remains open.

We study holomorphic and formal first integrals for germs of codimension-one holomorphic foliations on normal complex analytic spaces. In dimension two, under the assumption that the dual graph of the exceptional divisor of a resolution is a tree, we prove that the foliation admits a holomorphic first integral if and only if its leaves are closed outside the singular point and only finitely many leaves accumulate at that point. This extends a classical integrability theorem of Mattei and Moussu to singular ambient spaces. We also prove a holomorphic prolongation theorem for normal quotient germs admitting a smooth quasi-étale cover and a smooth connected lift of a generic two-dimensional section. We record, in addition, a conditional formal prolongation statement under depth assumptions on the conormal powers and an injectivity condition for the corresponding differential-form obstruction modules. Under the quotient-prolongation hypothesis, and with a reduced tangent cone where formal restriction must be detected, the higher-dimensional integrability results follow from their surface counterparts. We give a reduced nonnormal example satisfying both dynamical conditions but admitting no holomorphic first integral, showing that normality is essential. Our arguments combine resolution of singularities, holonomy techniques, formal completion, and extension properties of holomorphic functions on normal analytic spaces.

We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.

We study relative cohomology for logarithmic differential forms and meromorphic forms that are relatively closed with respect to the associated logarithmic foliation. Under suitable Diophantine, geometric, and topological hypotheses, we obtain decompositions into a meromorphic multiple of the defining logarithmic form, an exact meromorphic form, and a logarithmic form with constant residues. We establish a meromorphic extension theorem from suitable two-dimensional sections and prove local, polynomial, and homogeneous versions of the relative-cohomology decomposition; the global polynomial result is proved in arbitrary dimension by ambient leafwise continuation and meromorphic extension. Resolution of singularities, non-nodal saturation, and holonomy gluing are used to treat singular logarithmic foliations. As an application, we derive formal normal forms for analytic integrable deformations, including a several-component result in dimension two and a two-component extension in higher dimensions.

This paper focuses on coefficient problems for the class of starlike functions related to a cardioid region. We derive sharp bounds for the Hankel determinants of logarithmic coefficients and logarithmic inverse coefficients of order two. Additionally, we completely solve the extremal problem for the third-order Hermitian–Toeplitz determinant by providing its sharp upper and lower bounds. All results are supported by extremal examples demonstrating the sharpness of the obtained estimates.