aa r X i v : . [ m a t h . C V ] M a r A NEW LOOK AT KRZYZ’S CONJECTURE
SAMUEL L. KRUSHKAL
Abstract.
Recently the author has presented a new approach to solving extremal problemsof geometric function theory. It involves the Bers isomorphism theorem for Teichm¨ullerspaces of punctured Riemann surfaces.We show here that this approach, combined with quasiconformal theory, can be alsoapplied to nonvanishing holomorphic functions from H ∞ . In particular this gives a proof ofan old open Krzyz conjecture for such functions and of its generalizations.The unit ball H ∞ of H ∞ is naturally embedded into the universal Teichm¨uller space,and the functions f ∈ H ∞ are regarded as the Schwarzian derivatives of univalent functionsin the unit disk. Primary: 30C50, 30C55, 30H05; Secondary:30F60
Key words and phrases:
Nonvanishing holomorphic functions, Krzyz’s conjecture, Schwarzianderivative, Teichm¨uller spaces, Bers isomorphism theorem
1. INTRODUCTORY REMARKS AND RESULTS1.1. Nonvanishing holomorphic functions and Krzyz’s conjecture . Consider the circularrings A ρ = { ρ < | z | < } , ρ ≥ , and denote by H ( D , A ρ ) the collection of holomorphic functions f from the unit disk D into A ρ .Regarding the points of A ρ as the constant functions on D , one obtains an embedding of this ringas a subset in the space H ∞ of bounded holomorphic functions in D with sup-norm. By H ∞ wedenote the unit ball of this space.The collections H ( D , A ρ ) broaden monotonically as ρ ց H ∞ := H ( D , A ) = [ ρ H ( D , A ρ )of all nonvanishing H ∞ -functions. This class has been actively investigated in geometric functiontheory from the 1940s, in view of interesting deep features of nonvanishing (see, e.g., [9], [19]).In 1968, Krzyz [14] conjectured that for all functions from H ∞ the following bound | c n | ≤ /e (1)is valid for any n >
1, with equality only for the function κ ( z n ) and its pre and post rotationsabout the origin , where κ ( z ) = exp (cid:16) z − z + 1 (cid:17) = 1 e + 2 e z − e z + ... . (2)This conjecture has been investigated by many mathematicians, however it still remains open.The estimate (1) was established only for some initial coefficients c n including all n ≤ Date : April 1, 2020 (krzyzLook1.tex). [18], [20], [22], [23]). For developments related to this problem, see, e.g., [2], [10], [11], [12], [15],[16], [19], [21], [22]. . Put α A ρ := max {| f ′ (0) | : f ∈ H ( D , A ρ ) } (3)and take a universal holomorphic covering map κ ρ : D → A ρ , on which this maximal value of | f ′ (0) | is attained, i.e., | κ ′ ρ (0) | = α A ρ .Every function f ∈ H ( D , A ρ ) admits the factorization f ( z ) = κ ρ ◦ b f ( z ) , (4)where b f is a holomorphic map of the disk D into itself (hence, it also belongs to H ∞ ).In geometric function theory, such a relation is regarded as a subordination of functions f to κ ρ ;it has been investigated mostly for univalent covers κ ρ . Let k ρ ( z ) = c + c z + · · · + c n z n + . . . , | z | < . The existence of extremal functions maximizing the coefficient c n ( f ) on H ( D , A ρ ) follows fromcompactness of these classes in the weak topology determined by the locally uniform convergenceon D . The main result of this paper is the following theorem, which implies the proof of Krzyz’sconjecture.
Theorem 1 . For all f ∈ H ∞ and any n > , max | c n | = κ ′ (0) = 2 /e, (5) with equality only for the function κ ( z n ) and its compositions with pre and post rotations aboutthe origin. This theorem is extended to the spaces H ( D , A ρ ) with sufficiently small ρ . Theorem 2 . There is a number r , < r < , such that for any ρ < ρ every extremal function f maximizing | c n | on the corresponding class H ( D , A ρ ) is of the form f ( z ) = ǫ κ ρ ( ǫ z n ) (6) with | ǫ | = | ǫ | = 1 . In the case ρ >
0, we do not have an assertion on uniqueness of the covering map κ ρ on whichthe maximal value (3) is attained. . To prove Theorems 1 and 2, we apply a new approach in geometric function theory recentlypresented in [13]. It involves the Bers isomorphism theorem for Teichm¨uller spaces of puncturedRiemann surfaces.The unit ball H ∞ of H ∞ is naturally embedded into the universal Teichm¨uller space T , and thefunctions f ∈ H ∞ are regarded as the Schwarzian derivatives of univalent functions in the unitdisk.In fact, this approach allows one to consider also some more general homogeneous polynomialcoefficient functionals than c n ( f ).
2. DIGRESSION TO TEICHM ¨ULLER SPACES Weak compactness of H ∞ is obtained after adding to this class the function f ( z ) ≡ new look at Krzyz’s conjecture 3 We briefly recall some needed results from Teichm¨uller space theory on spaces in order to proveTheorem 1; the details can be found, for example, in [3], [8]. . The universal Teichm¨uller space T = Teich( D ) is the space of quasisymmetric homeomor-phisms of the unit circle S factorized by M¨obius maps; all Teichm¨uller spaces have their isometriccopies in T .The canonical complex Banach structure on T is defined by factorization of the ball of theBeltrami coefficients (or complex dilatations) Belt ( D ) = { µ ∈ L ∞ ( C ) : µ | D ∗ = 0 , k µ k < } , letting µ , µ ∈ Belt ( D ) be equivalent if the corresponding quasiconformal maps w µ , w µ (solu-tions to the Beltrami equation ∂ z w = µ∂ z w with µ = µ , µ ) coincide on the unit circle S = ∂ D ∗ (hence, on D ∗ ). Such µ and the corresponding maps w µ are called T - equivalent . The equivalenceclasses [ w µ ] T are in one-to-one correspondence with the Schwarzian derivatives S w ( z ) = (cid:18) w ′′ ( z ) w ′ ( z ) (cid:19) ′ − (cid:18) w ′′ ( z ) w ′ ( z ) (cid:19) ( w = w µ ( z ) , z ∈ D ∗ ) . Note that for each locally univalent function w ( z ) on a simply connected hyperbolic domain D ⊂ b C , its Schwarzian derivative belongs to the complex Banach space B ( D ) of hyperbolicallybounded holomorphic functions on D with the norm k ϕ k B = sup D λ − D ( z ) | ϕ ( z ) | , where λ D ( z ) | dz | is the hyperbolic metric on D of Gaussian curvature −
4; hence ϕ ( z ) = O ( z − ) as z → ∞ if ∞ ∈ D . In particular, for the unit disk, λ D ( z ) = 1 / (1 − | z | ) . The space B ( D ) is dual to the Bergman space A ( D ), a subspace of L ( D ) formed by integrableholomorphic functions (quadratic differentials ϕ ( z ) dz on D ), since every linear functional l ( ϕ ) on A ( D ) is represented in the form l ( ϕ ) = h ψ, ϕ i D = Z Z D λ − D ( z ) ψ ( z ) ϕ ( z ) dxdy (7)with a uniquely determined ψ ∈ B ( D ).The Schwarzians S w µ ( z ) with µ ∈ Belt ( D ) range over a bounded domain in the space B = B ( D ∗ ). This domain models the space T . It lies in the ball {k ϕ k B < } and contains the ball {k ϕ k B < } . In this model, the Teichm¨uller spaces of all hyperbolic Riemann surfaces are containedin T as its complex submanifolds.The factorizing projection φ T ( µ ) = S w µ : Belt ( D ) → T is a holomorphic map from L ∞ ( D ) to B . This map is a split submersion, which means that φ T haslocal holomorphic sections (see, e.g., [GL]).Note that both equations S w = ϕ and ∂ z w = µ∂ z w (on D ∗ and D , respectively) determine theirsolutions in Σ θ uniquely, so the values w µ ( z ) for any fixed z ∈ C and the Taylor coefficients b , b , . . . of w µ ∈ Σ θ depend holomorphically on µ ∈ Belt ( D ) and on S w µ ∈ T . . The points of Teichm¨uller space T = Teich( D ∗ ) of the punctured disk D ∗ = D \ { } arethe classes [ µ ] T of T - equivalent Beltrami coefficients µ ∈ Belt ( D ) so that the correspondingquasiconformal automorphisms w µ of the unit disk coincide on both boundary components (unitcircle S = {| z | = 1 } and the puncture z = 0) and are homotopic on D \ { } . This space can beendowed with a canonical complex structure of a complex Banach manifold and embedded into T using uniformization. Samuel L. Krushkal
Namely, the disk D ∗ is conformally equivalent to the factor D / Γ, where Γ is a cyclic parabolicFuchsian group acting discontinuously on D and D ∗ . The functions µ ∈ L ∞ ( D ) are lifted to D as theBeltrami ( − , e µdz/dz in D with respect to Γ, i.e., via ( e µ ◦ γ ) γ ′ /γ ′ = e µ, γ ∈ Γ,forming the Banach space L ∞ ( D , Γ).We extend these e µ by zero to D ∗ and consider the unit ball Belt ( D , Γ) of L ∞ ( D , Γ). Thenthe corresponding Schwarzians S w e µ | D ∗ belong to T . Moreover, T is canonically isomorphic to thesubspace T (Γ) = T ∩ B (Γ), where B (Γ) consists of elements ϕ ∈ B satisfying ( ϕ ◦ γ )( γ ′ ) = ϕ in D ∗ for all γ ∈ Γ.Due to the Bers isomorphism theorem, the space T is biholomorphically isomorphic to the Bersfiber space F ( T ) = { ( φ T ( µ ) , z ) ∈ T × C : µ ∈ Belt ( D ) , z ∈ w µ ( D ) } over the universal space T with holomorphic projection π ( ψ, z ) = ψ (see [3]).This fiber space is a bounded hyperbolic domain in B × C and represents the collection of domains D µ = w µ ( D ) as a holomorphic family over the space T . For every z ∈ D , its orbit w µ ( z ) in T is aholomorphic curve over T .The indicated isomorphism between T and F ( T ) is induced by the inclusion map j : D ∗ ֒ → D forgetting the puncture at the origin via µ ( S w µ , w µ (0)) with µ = j ∗ µ := ( µ ◦ j ) j ′ /j ′ , (8)where j is the lift of j to D .In the line with our goals, we slightly modified the Bers construction, applying quasiconformalmaps F µ of D ∗ admitting conformal extension to D ∗ (and accordingly using the Beltrami coefficients µ supported in the disk) (cf. [13]). These changes are not essential and do not affect the underlyingfeatures of the Bers isomorphism (giving the same space up to a biholomorphic isomorphism).The Bers theorem is valid for Teichm¨uller spaces T ( X \ { x } ) of all punctured hyperbolicRiemann surfaces X \ { x } and implies that T ( X \ { x } ) is biholomorphically isomorphic to theBers fiber space F ( T ( X )) over T ( X ).Note that B (Γ ) has the same elements as the space A ( D ∗ , Γ ) of integrable holomorphic forms ofdegree − k ϕ k A ( D ∗ , Γ ) = RR D ∗ / Γ | ϕ ( z ) | dxdy ; and similar to (10), every linear functional l ( ϕ ) on A ( D ∗ , Γ ) is represented in the form l ( ϕ ) = h ψ, ϕ i D / Γ := Z Z D ∗ / Γ (1 − | z | ) ψ ( z ) ϕ ( z ) dxdy with uniquely determined ψ ∈ B (Γ ).Any Teichm¨uller space is a complete metric space with intrinsic Teichm¨uller metric definedby quasiconformal maps. By the Royden-Gardiner theorem, this metric equals the hyperbolicKobayashi metric determined by the complex structure (see, e.g., [7], [8]).We do not use here the finite dimensional Teichm¨uller spaces corresponding to finitely generatedFuchsian groups.
3. PROOF OF THEOREM 1
We carry out the proof in several stages as a consequence of lemmas. . Let G be a ring subdomain of the disk D bounded by the unit circle S = ∂ D and a Jordancurve γ G separating the origin and S , and let H ( D , G ) denote the subspace of functions f ∈ H ∞ mapping D into G .We first establish some results characterizing the structure of such sets H ( D , G ). new look at Krzyz’s conjecture 5 Lemma 1 . Any set H ( D , G ) contains an open path-wise connective subdomain H ( D , G ) whichis dense in the weak topology of locally uniform convergence on D , and the universal holomorphiccovering map κ G : D → G extends to a holomorphic map from H ∞ onto H ( D , G ) (that is, in H ∞ -norm). Proof . We precede the proof of this main lemma by two auxiliary lemmas giving other analyticand geometric features of sets H ∞ ( D , G ). Lemma 2 . (a) Every function f ∈ H ( D , G ) admits factorization f ( z ) = κ G ◦ b f ( z ) , (9) where b f is a holomorphic map of the disk D into itself (hence, from H ∞ ).(b) Moreover, the relation generates an H ∞ -holomorphic map k G : b f f from H ∞ onto H ( D , G ) . Proof . (a) Due to a general topological theorem, any map f : M → N , where M, N are manifolds,can be lifted to a covering manifold b N of N , under an appropriate relation between the fundamentalgroup π ( M ) and a normal subgroup of π ( N ) defining the covering b N (see, e.g, [17]). Thisconstruction produces a map b f : M → b N satisfying f = p ◦ b f , (10)where p is a projection b N → N . The map b f is determined up to composition with the coveringtransformations of b N over N or equivalently, up to choosing a preimage of a fixed point x ∈ b N inits fiber p − ( x ). For holomorphic maps and manifolds, the lifted map is also holomorphic.In our special case, κ G is a holomorphic universal covering map D → G , and the representation(10) provides the equality (9) with the corresponding b f determined up to covering transformationsof the unit disk compatible with the covering map κ G .For a fixed z ∈ D , each coefficient c n , of f (and hence f itself) is a holomorphic function(polynomial) of the initial coefficients b c , b c , . . . , b c n of cover b f . Holomorphy in the H ∞ norm statedby the assertion (b) is a consequence of a well-known property of bounded holomorphic functionsin Banach spaces with sup norm given by the following lemma of Earle [6]. Lemma 3 . Let
E, T be open subsets of complex Banach spaces
X, Y and B ( E ) be a Banach spaceof holomorphic functions on E with sup norm. If ϕ ( x, t ) is a bounded map E × T → B ( E ) suchthat t ϕ ( x, t ) is holomorphic for each x ∈ E , then the map ϕ is holomorphic. Holomorphy of ϕ ( x, t ) in t for fixed x implies the existence of complex directional derivatives ϕ ′ t ( x, t ) = lim ζ → ϕ ( x, t + ζv ) − ϕ ( x, t ) ζ = 12 πi Z | ξ | =1 ϕ ( x, t + ξv ) ξ dξ, while the boundedness of ϕ in sup norm provides the uniform estimate k ϕ ( x, t + cζv ) − ϕ ( x, t ) − ϕ ′ t ( x, t ) cv k B ( E ) ≤ M | c | , for sufficiently small | c | and k v k Y .The map k ρ : b f f is bounded on the ball H ∞ . Applying Hartog’s theorem on separateholomorphy to the sums g ( z, t ) = b f ( z )+ t b h ( z ) of b f ∈ H ∞ , b h ∈ H and t from a region B ⊂ b C so that g ( z, t ) ∈ H ∞ , one obtains that g ( z, t ) are jointly holomorphic in both variables ( z, t ) ∈ D × B . Thusthe restriction of the map k G onto intersection of the ball H ∞ with any complex line L = { b f + t b h } is H ∞ -holomorphic, and hence this map is holomorphic as the map H ∞ → H ( D , G ), which completesthe proof of Lemma 3. Samuel L. Krushkal
One can also show that the restriction of the extended map k G to any holomorphic disk D ( b f ) = { t b f / k b f k ∞ : | t | < } , b f ∈ H ∞ , is a complex geodesic (cf. [25]), hence a local hyperbolic isometry (preserving such property of theoriginal map k G ). We will not use this fact and therefore do not present here its proof.Now consider the domains G ⋐ D . Fix a point w ∈ G and take for a decreasing sequence r n → G n of widening open sets G n = { w ∈ G : dist( w, ∂G ) > r n } , G n ⋐ G n +1 , n = 1 , , . . . , exhausting this domain and containing w . Let H ( D , G ) = [ n H ( D , G n ) . The set H ( D , G ) is open and contains, in particular, all functions f ∈ H ( D , G ) holomorphic onthe closed disk D . Lemma 4 . (a) For any fixed n , every function f ∈ H ( D , G n ) continuous in the the closed disk D has a neighborhood U ( f, ǫ n ) in H ∞ , which contains only the functions belonging to H ( D , G ) .(b) Each of the sets H ( D , G n ) and H ( D , G ) is path-wise connective in H ∞ ; therefore, the union H ( D , G ) is a domain in H ∞ . Proof . To prove the assertion (a) , assume to the contrary that, for some n = n , such a number m ( n ) does not exist. Then there is a function f ∈ H ( D , G n ), a sequence of functions f m ∈ H ( D , G n ) convergent to f so that lim m →∞ k f m − f k H ∞ = 0 , (11)and a sequence of points z m ∈ D convergent to z ∈ D , for which we will have f m ( z m ) ∈ G m for all m ≥ n , and lim m →∞ f m ( z m ) = a ∈ D \ G. (12)We approximate f m ( z ) by functions f m,r ( z ) = f m ( r m z ) (holomorphic in D ), taking r m so closeto 1 that the equality (11) is preserved for f m,r . Then the uniform convergence of f m and f m,r to f on compact subsets of D immediately implies that the limit a in (15) must be equal to f ( z ),and therefore it belongs to G n . This proves part (a) .To show that each H ( D , G n ) is path-wise connective, take its arbitrary distinct points f , f .By (9), f j = κ G n ◦ e f j ( e f j ∈ H ∞ ) , j = 1 , . Connecting the covers e f and e f in H ∞ by the line interval l , ( t ) = t e f + (1 − t ) e f ≤ t ≤ κ G n ◦ l , : [0 , → H ( D , G n ) connecting f with f , completing the proof ofLemmas 4 and 1.Observe that Lemma 4 does not contradict to existence for f ∈ H ( D , G ) of sequences { f n } ∈ H ∞ convergent to f only locally uniformly in D and taking some values in D \ G .Using the homotopy b f t ( z ) = t b f ( tz ) of the cover functions and representation (9), one concludesthat the domain H ( D , G ) is dense in the set H ( D , G ) in the weak topology. Hence,sup H ( D ,G ) | J ( f ) | = max H ( D ,G ) | J ( f ) | for any holomorphic functional J ( f ). This follows also from the fact that all f ∈ H ( D , G ) holo-morphic on the closure G of domain G belong to H ( D , G ). new look at Krzyz’s conjecture 7 Note also that for G = A ρ , the distinguished domain H ( D , A ρ ) preserves circular symmetry,i.e., it contains the nonvanishing functions f ∈ H ∗ together with their compositions with pre andpost rotations about the origin. . In the case of the punctured disk D ∗ = A Lemma 4 admits some strengthening.
Lemma 5 . Each point f ∈ H ∞ has a neighborhood (ball) U ( f, ǫ ) in H ∞ , which entirely belongs to B , i.e., contains only nonvanishing functions on the disk D . Take the maximal balls U ( f, ǫ ) withsuch property. Then their union U = [ f ∈ H ∞ U ( f, ǫ ) is a domain in the space H ∞ . Proof . Openness : It suffices to show that for each r > r ′ < r , every function f ∈ H ∞ has aneighborhood U ( f, ǫ ( r )) in H ∞ ( D r ′ ), which contains only nonvanishing functions on D r ′ = {| z |
12 ( | ζ | − ζ ζ S w (cid:16) ζ (cid:17) . We shall denote this holomorphic embedding of the ball H ∞ into the space T modeled by Schwarziansin D ∗ again by ι . The image of H ∞ under this embedding is a noncomplete linear subspace in B so that ιH ∞ is a complex subset of the unit ball in B , and the image of the distinguished domain H ( D , A ρ ) is a complex submanifold in T .Another important property of the set ιH ( D , A ρ ) is given by the following two lemmas. Lemma 6 . Let f ( z ) = ∞ P c n z n ∈ H ∞ and s m ( z ) = m − P c n z n . Then lim m →∞ k s m − f k B = 0 . (16) Proof . It suffices to consider the functions f from the ball {k f k B < / } . Their coefficients c j areestimated by | c n | < / n ≥
0. Hence, | s m ( z ) | < m − X | z | n < − | z | ) , and k s m k B <
12 sup D (1 − | z | )(1 + | z | ) < , which means that any partial sum s n for such f lies in the ball { g ∈ B : k g k B < } and thereforeit also belongs to the space T . Further, | s m ( z ) − f ( z ) | = | c m z m + c m +1 z m +1 + . . . | <
12 ( | z | m + | z | m +1 + . . . )= 12 | z | m − | z | < m +1 − | z | , which implies k s m ( z ) − f ( z ) k B < m +1 sup D (1 − | z | )(1 + | z | ) < m ‘ − , and (19) follows, proving the lemma.Lemmas 1 and 5 imply Lemma 7 . Any point f ( z ) = P ∞ c n z n from the set ιH ( D , A ρ ) in T is approximated in the B -norm by polynomials s m ( z ) = c + c z + · · · + c m z m with m ≥ m ( f ) , which also belong to ιH ( D , A ρ ) and hence do not have zeros in the disk D . . Our next step is to lift both polynomial functionals J n ( w ) and b J n ( W ) (equivalently c n ) ontothe Teichm¨uller space T . Letting b J n ( µ ) = e J n ( W µ ) , (17)we lift these functionals from the sets S θ (1) and Σ θ (1) onto the ball Belt ( D ) . Then, under theindicated T -equivalence, i.e., by the quotient map φ T : Belt ( D ) → T , µ → [ µ ] T , the functional e J n ( W µ ) is pushed down to a bounded holomorphic functional J n on the space T with the same range domain. Equivalently, one can apply the quotient map
Belt ( D ) → T (i.e., T -equivalence) and composethe descended functional on T with the natural holomorphic map ι : T → T generated by theinclusion D ∗ ֒ → D forgetting the puncture. Note that since the coefficients b , b , . . . of W µ ∈ Σ θ are uniquely determined by its Schwarzian S W µ , the values of J n in the points X , X ∈ T with ι ( X ) = ι ( X ) are equal.Now, using the Bers isomorphism theorem, we regard the points of the space T as the pairs X W µ = ( S W µ , W µ (0)), where µ ∈ Belt ( D ) obey T -equivalence (hence, also T -equivalence).Denote (for simplicity of notations) the composition of J n with biholomorphism T ∼ = F ( T ) againby J n . In view of (8) and (17), it is presented on the fiber space F ( T ) by J n ( X W µ ) = J n ( S W µ , t ) , t = W µ (0) . (18)This yields a logarithmically plurisubharmonic functional |J n ( S W µ , t ) | on F ( T ).Note that since the coefficients b , b , . . . of W µ ∈ Σ θ are uniquely determined by its Schwarzian S W µ , the values of J n in the points X , X ∈ T with ι ( X ) = ι ( X ) are equal.We have to estimate a smaller plurisubharmonic functional arising after restriction of J n ( S F µ , t )onto the the images in these spaces of the distinguished convex set ιH ( D , A ρ ), i.e., the functional(17) on the set of S W µ ∈ ιH ( D , A ρ ) and corresponding values of t = W µ (0) which runs over somesubdomain D ρ,θ in the disk {| t | < } .We denote this restricted functional by J n, ( S W µ , W µ (0)) and define in domain D ρ,θ the function u θ ( t ) = sup S Wµ |J n, ( S W µ , t ) | , (19)where the supremum is taken over all S F µ ∈ ιH ( D , A ρ ) admissible for a given t = W µ (0) ∈ D ρ,θ ,that means over the pairs ( S W µ , t ) ∈ F ( T ) with S F µ ∈ ιH ( D , A ρ ) and a fixed t .Our goal is to establish that this function inherits the subharmonicity of J . This is given by thefollowing basic lemma. Lemma 8 . The function u θ ( t ) is subharmonic in the domain D ρ,θ . Proof . Consider in the set ιH ( D , A ρ ) its m -dimensional analytic subsets V m corresponding to thepartial sums s m of functions f ∈ H ( D , A ρ ) (with m ≥ m ( f )). Given such f , we define F ( z ) = f (1 /z ) /z and take a univalent solution W ∈ Σ θ of the Schwarzian equation S W ( z ) = F ( z ) on D ∗ . Let W µ be one of its quasiconformal extensions onto D .Let W m and W µ m m be the corresponding functions defined similarly by the partial sums s m of f, m ≥ m ( f ). Then the domains W m ( D ∗ ) and W µ m m ( D ) approximate W ( D ∗ ) and W µ ( D ) uniformly(in the spherical metric on b C ), and the points W µ m n (0) are close to W µ (0).One can replace the extensions W µ m n by ω m ◦ W µ m n , where ω m is the extremal quasiconformalautomorphism of domain W µ m m ( D ) moving the point W µ m m (0) into W µ (0) and identical on theboundary of W µ m m ( D ) (cf. [24]). This provides for a prescribed t = W µ (0) the points S W µmm ∈ F ( T )corresponding to given s m ∈ V m .Now, maximizing the function log |J n, ( S W µmm , t ) | over the manifold V m , i.e., over S W µmm (withappropriate m ), one obtains a logarithmically plurisubharmonic function u m ( t ) = sup V m |J n, ( S W µmm , t ) | , t = W µ (0) , in the domain D ρ,θ indicated above. We take its upper semicontinuous regularization u m ( t ) = lim sup t ′ → t u m ( t ′ ) new look at Krzyz’s conjecture 11 (denoted, by abuse of notation, by the same letter as the original function). The general prop-erties of subharmonic functions in the Euclidean spaces imply that such a regularization also islogarithmically subharmonic in D ρ,θ .In a similar way, taking the limit u ( t ) = lim sup m →∞ u m ( t )followed by its upper semicontinuous regularization, one obtains a logarithmically subharmonicfunction on the domain D ρ,θ . Lemmas 6 and 7 imply that this function coincides with function(19). . Assume now that ρ = 0, hence A ρ = D ∗ = D \ { } .We have to establish the value domain of W µ (0) for W µ running over ιH ( D , A ).First, we apply the following generalization of the above construction. Taking a dense countablesubset Θ = { θ , θ , . . . , θ m , . . . } ⊂ [ − π, π ] , consider the increasing unions of the quotient spaces T m = m [ j =1 b Σ θ j / ∼ = m [ j =1 { ( S W θj , W µθ (0)) } ≃ T ∪ · · · ∪ T , (20)where the equivalence relation ∼ means T -equivalence on a dense subset b Σ (1) in the union b Σ(1)formed by all univalent functions W θ ( z ) = e − iθ j z + b + b z − + . . . on D ∗ (preserving z = 1) withquasiconformal extension to b C , and W µθ (0) := ( W µ θ (0) , . . . , W µ m θ m (0)) . The Beltrami coefficients µ j ∈ Belt ( D ) are chosen here independently. The corresponding collec-tion β = ( β , . . . , β m ) of the Bers isomorphisms β j : { ( S W θj , W µ j θ j (0)) } → F ( T )determines a holomorphic surjection of the space T m onto F ( T ).Taking also in each union (20) the corresponding collection ι m H ∞ covering H ( D , A ), oneobtains in a similar manner to the above the maximal function u ( t ) = sup Θ u θ m ( t ) = sup {|J n, ( S W µθ , t ) | : θ ∈ [ m ι m H ∞ } . (21)It is defined and subharmonic in domain D ρ = [ Θ D ρ,θ m . Noting that the union of spaces T m possesses the circular symmetry inherited from the class b Σ(1), which is preserved under rotations (14), one concludes that this broad domain D must be adisk D r = {| t | < r } .Now we show that in the case of nonvanishing H ∞ functions this radius r is naturally connectedwith the function (2). This requires a covering estimate of Koebe’s type.Let G be a domain in a complex Banach space X = { x } and χ be a holomorphic map from G into the universal Teichm¨uller space T modeled as a bounded subdomain of B . Consider in theunit disk the corresponding Schwarzian differential equations S w ( z ) = χ ( x ) (22)and pick their univalent solutions w ( z ) satisfying w (0) = w ′ (0) − w ( z ) = z + P ∞ a n z n ).Put | a | = sup {| a | : S w ∈ χ ( G ) } , (23) and let w ( z ) = z + a z + . . . be one of the maximizing functions. Lemma 10 . (a) For every indicated solution w ( z ) = z + a + . . . of (22), the image domain w ( D ) covers entirely the disk {| w | < / (2 | a | ) } .The radius value / (2 | a | ) is sharp for this collection of functions, and the circle {| w | = 1 / (2 | a | ) contains points not belonging to w ( D ) if and only if | a | = | a | (i.e., when w is one of the maximizingfunctions).(b) The inverted functions W ( ζ ) = 1 /w (1 /ζ ) = ζ − a + b ζ − + b z − + . . . map the disk D ∗ onto a domain whose boundary is entirely contained in the disk {| W + a | ≤ | a |} . The proof follows the classical lines of Koebe’s 1 / (a) Suppose that the point w = c does not belong to the image of D under the map w ( z ) definedabove. Then c = 0, and the function w ( z ) = cw ( z ) / ( c − w ( z )) = z + ( a + 1 /c ) z + . . . also belongs to this class, and hence by (23), | a + 1 /c | ≤ | a | , which implies | c | ≥ / (2 | a | ) . The equality holds only when | a + 1 /c | = | /c | − | a | = | a | and | a | = | a | . (b) If a point ζ = c does not belong to the image W ( D ∗ ), then the function W ( z ) = 1 / [ W (1 /z ) − c ] = z + ( c + a ) z + . . . is holomorphic and univalent in the disk D , and therefore, | c + a | ≤ | a | . The lemma follows.This lemma implies that the boundary of the range domain of W µ (0) is contained in the disk D | a | = { W : | W | ≤ | a |} , (24)and, consequently, r = 2 | a | and touches from inside the circle {| W | = 2 | a |} at the pointscorresponding to extremal functions W maximizing | a | on the closure of the domain ιH ∗ .Generically, the extremal value 2 | a | of the radius of covered disk can be attained on severalfunctions W . . We now establish that S W ( z ) = κ ( z ) . (25)In view of Lemma 1, it is enough to show that S ′ W (0)( z ) = c = 0 (26)(in other words, that the zero set of the functional J ( f ) = c is separated from the set of rota-tions (14) of the function W ). This yields that the correspomdimg function (21), constructed bymaximization of functional J ( f ) = | c | , is defined and subharmonic on the whole disk D | a | ), andits maximaum is attained on the boundary circle.Assume, to the contrary, that S ′ W (0)( z ) = 0. Then, by Lemma 2, S W ( z ) = κ ◦ b f ( z ) = c + c z + c z + . . . , where b f is a holomorphic self-map of D of the form b f ( z ) = b c + b c z + b c z + . . . . new look at Krzyz’s conjecture 13 Since the function κ ( z ) = κ ( z ) = 1 /e + (2 /e ) z + . . . also belongs to ιH ∗ , it must be | c | > /e. (27)Now consider the function b f ( z ) = σ − ◦ n b f ◦ σ ( z ) z o = b c + b c z + b c z + . . . , where σ ( z ) = ( z − b c ) / (1 − b c z ) . This function also is a holomorphic self-map of the disk D . Its composition with κ via (10),denoted by f , is a nonvanishing holomorphic self-map of D , and a simple calculation, using (27),yields f ′ (0) = ( κ ◦ b f ) ′ (0) = | c | > /e, which contradicts to Lemma 1. This proves the relations (25) and (26). . Now we can finish the proof of the theorem.Take n = 2 and, letting f ( z ) = f ( z ), consider on H ∞ the plurisubharmonic functional I ( f ) = max ( | J ( f ) | , | J ( f ) | ) . (28)Similar to above, the lift of this functional onto T generates via (19) a nonconstant radial subhar-monic function of on the disk (24). It is logarithmically convex, hence monotone increasing, andattains its maximal value at | t | = 2 | a | .By Parseval’s equality for the boundary functions f ( e iθ ) = lim r → f ( re iθ ) of f ∈ H ∞ , we have1 ≥ π π Z π | f ( e iθ ) | dθ = ∞ X | c n | . Applying it to the function f ( z ) = κ ( z ) = ∞ X c n z n and noting that by (2), | c | = 4 e − = 0 . ... , one obtains that for this function, ∞ X | c n | < . < | c | . This implies (in view of the indicated connection of | a | with κ ) that the maximal value of thefunctional (28) on H ∞ is attained on the functions κ ( z ) , κ ( z ) = κ ( z ) and equalsmax ( | c | , | c | ) = 2 /e, giving the desired estimate (1) for n = 2. The extremal extremal function is unique, up to rotations.Now take n = 3 and, letting f ( z ) = f ( z ), consider similar to (28) the functional I ( f ) = max ( | J ( f ) | , | J ( f ) | )Arguing similar to the above case, one obtainsmax H ∞ I ( f ) = max ( | c | , | c | ) = 2 /e, giving the estimate (1) for n = 3.Taking subsequently n = 4 , , . . . , one obtains by the same arguments that the estimate (1) isvalid for all n , completing the proof of Theorem 1.
4. PROOF OF THEOREM 2
In view of the uniform convergence of H ∞ functions on compact subsets of the unit disk, we havefor the covering maps κ ρ : D → A ρ , that their derivatives κ ′ ρ (0) are convergent to κ ′ (0) = 2 /e < ρ → ρ < ρ so small that κ ′ ρ (0) <
1, one can repeat the above arguments applied in the proofof Theorem 1 to the corresponding sets H ( D , A ρ ) with such ρ .But now we do not have the assertion on uniqueness of the covering map κ ρ on which the maximalvalue (3) is attained.
5. REMARK ON THE HUMMEL-SCHEINBERG-ZALCMAN CONJECTURE
The Krzyz conjecture was extended in 1977 by Hummel, Scheinberg and Zalcman to arbitraryHardy spaces H p , p > f ( z ) ∈ H p , p >
1, with k f k p ≤ | c n | ≤ (2 /e ) − /p , with equality for the function f n ( z ) = h (1 + z n ) i /p h exp z n − z n + 1 i − /p and its rotations (see [11]). As p → ∞ , this yields Krzyz’s conjecture for H ∞ (without uniquenessof extremal functions)This problem also has been investigated by many authors, but it still remains open. The onlyknown results here are that the conjecture is true for n = 1 proved by Brown [4] as well as someresults for special subclasses of H p , see [4], [5], [21].Some important intrinsic features of H ∞ functions, essentially involved in the proof of Krzyz’sconjecture, are lost in H p . However, the above arguments can be appropriately modified andcompleted to include also the Hummel-Scheinberg-Zalcman conjecture. This will be presented in aseparate work. References [1] J. Agler and J. McCarthy,
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