On Gol dberg s constant A 2

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1 On Gol dberg s constant A 2 Prashant Batra 4th June 2006 Abstract Gol dberg considered the class of functions analytic in the unit disc with unequal positive numbers of zeros and ones there. The maximum modulus of zero- and one-places in this class is non-trivially bounded from below by the universal constant A 2. This constant determines a fundamental limit of controller design in engineering, and has applications when estimating covering regions for composites of fixed point free functions with schlicht functions. The lower bound for A 2 is improved in this note by considering simultaneously the extremal functions f and f together with their reciprocals. Keywords: Gol dberg s second constant, Schottky s theorem, Borel-Hadamard inequalities, value distribution, holomorphic functions AMS No. 30C5, 93D5 Introduction Landau showed in [] that there exists a universal constant R = Ra 0, a ) such that all functions fz) = a 0 +a z+... analytic in the disc of radius R must attain one of the values 0 and in this disc. This result was generalised by Schottky who showed in [5] that for an analytic function fz) which omits 0 and on z = r < there exists a modulus limit depending merely on f0) and r. It was Gol dberg [6] who discussed the minimum maximum modulus problem of zero- and one-places under the general form considered in this note. Given a function f holomorphic inside the unit disc D. Let n 0 := {z D : fz) = 0}, n := {z D : fz) = }. Denote the class of holomorphic functions f on D such that 0 n 0 n 0 by K 2. Denote by rf) the maximum modulus of the zero- and one-places of f K 2. Denote by A 2 the greatest lower bound for all rf) with f K 2. Gol dberg [6] showed that the universal constant A 2 is strictly positive, and proved the estimate A 2 >

2 An example is used in [6] to establish an upper bound to A 2. Gol dberg claims that the example provides the upper bound 0.03 > A 2. Related results of Jenkins [0] imply that A 2 > , while Blondel, Rupp and Shapiro [3] established rf) > exp + 2 ) π e )π2, where N := Nf) := min{n 0, n }. ) N The bound ) improves on Jenkins bound for given f whenever f is such that N 5. The author established recently [] that A 2 > The quest for narrow bounds on the universal constant A 2 can be motivated by mathematical as well as engineering consequences. As Rupp [4] showed, the composite of a normalized schlicht function with a fixed point free function covers a disc dependent in radius on Gol dberg s constant A 2. This quantity is also of interest in the engineering context of stabilization. The existence of certain stabilizing, time-invariant controllers i.e. meromorphic functions) may be shown to be equivalent to the existence of a function with prescribed zeros and ones in a generalized circle. Several benchmark problems were posed by Blondel in [4] relating controller design to the value distribution of functions. For a reformulation of the control question in terms of the value distribution of functions, see [3]. The known lower bound to A 2 due to Gol dberg as well as the one in [3] rely on quantitative maximum modulus estimates in Schottky s theorem in combination with the Poisson-Jensen formula. In this note, we deviate from this approach combining maximum modulus estimates with minimum modulus estimates on parametrized circles. The outline of this note is as follows. After a short overview on quantitative estimates for Schottky s theorem, a mapping κ from D to the annulus r < z < where r = rf) is the maximum modulus of zero- and one-places of f K 2 ) is outlined. Using the composition of f K 2 with this mapping fz) and fz) are estimated on z = r taking the following steps. On every circle of radius ρ, where r < ρ < r, there exists at least one point z such that at least one of the functions f, f K 2 takes a real value not exceeding /2. The zeros of f give rise to a Blaschke product B unimodular on a circle of radius no less than r. The Hadamard-Borel-Carathéodory inequalities then allow us to estimate from above the minimum modulus of f on z = r considering /f B) at the point z. 2

3 This yields a function value smaller in modulus than /2 on the outer radius r. Zhang s sharp, quantitative version of Schottky s theorem gives an improved maximum modulus estimate for fκ )), and hence for f inside an annulus. The maximum modulus bound can be used to estimate the minimum modulus from below via the Hadamard-Borel-Carathéodory inequalities for the product of f with the zero-cancelling Blaschke product B. The lower bound for f and f must be small as can be seen by an application of the argument principle. A proper choice of parameters allows us to conclude that A 2 > Magnitude estimates via Schottky s theorem Quantitative versions of Schottky s theorem have been sought for a long time. Schottky [5] established the following fundamental result. Theorem Let gz) = a 0 + a z + a 2 z be regular in z < r and unequal to 0 or in this circle. Then ) 4 lngz)) < 224 r, α r z { } where α := min ln a 0, ln a 0 ), ln a 0 a 0. Hayman [7] showed that under the assumptions of the preceeding theorem gz) < 6 µeπ ) +r r for all z with z < r, where µ := max{ a 0, }, and moreover, that π may not be replaced by a smaller constant. Ostrowski [2] had shown earlier that + r)/ r) is the precise asymptotic order for a 0. Several different ways to obtain estimates dependent on a 0 were pursued by Hayman, Jenkins, Lai, Hempel and Zhang cf. [8] for references). As we wish to apply the modulus estimate to f and f simultaneously, we consider Zhang s version [6] of Schottky s theorem which takes a 0 into account even for small modulus and slightly improves on [9]). Theorem 2 Let g be a holomorphic function without zeros and ones inside the unit disc. Then gz) Ka, z ), for all z D, 2) where Ka, z ) := 6 exp π 2 + z ) ln6/a + b) z ) ), a := g0), b := eπ 6. In case that K = Ka, z ) >, the right-hand side of inequality 2) can be replaced by Ka, z ) b/6. 3

4 Suppose the maximum modulus of the zero- and one-places of f is r. To employ Schottky s theorem it is useful to map the unit disc to the annulus A := {z C : r < z < } where f and f are zero-free. Following Gol dberg see also [3]) we map D onto the annulus as follows. Map the unit disc via the logarithmic function ηz) = i ln i +z to the infinite vertical strip S of width π centered at π/2. The z strip S is mapped onto the annulus A by the power function νξ) = r ξ/π). By concatenation obtain the surjective mapping κ : D A, κz) := νηz)). The segment π/2 + λi with π 2 / lnr) λ π 2 / lnr) is mapped by ν to the circle z = r. The pre-image of the segment point ξ 0 = π/2 + λi S under η has maximum modulus tanh λ/2. Hence, there exists a preimage of the curve z = r D under the map κ bounded in modulus see [6], p.205, or [3], p.90, eqs.)-5)) by tanhπ 2 /2 lnr) ). 3) Compose κ with a function f unequal to zero or one in the annulus A to obtain a mapping g ) := fκ )) satisfying the assumptions of Theorem 2. Modulus estimates for g0) = fκ0)) as required for numerical computations involving Theorem 2 will be derived in the next section. 3 Limits to Minimum and Maximum Modulus A function f in K 2 has different positive winding numbers around 0 and on a circle z = ρ, rf) < ρ <. Hence, with r := rf), the image of every circle z = ρ, r < ρ < under f crosses the section between 0 and. This gives min { fz), fz) } <. 4) z =ρ,r<ρ< Moreover, on any circle z = ρ, rf) = r < ρ < we may choose dependent on ρ) f ρ as one of the functions f and f such that ρ r,) z=ρe iφ fρ {f, f} 0 < f ρ z) 2. 5) Especially for ρ = r we obtain z with z = ρ and f ρ z) /2. This estimate is improved for r = using a vital source of Schottky s theorem: functions zero-free and bounded in a disc z < r can be estimated on an interior circle z = ρ, ρ < r =: r ext using Carathéodory s improved formulation first in [5]) of the Borel-Hadamard inequalities [3]. 4

5 Lemma For a holomorphic function h unequal to zero in the disc z < r ext, and bounded there in modulus by M h we have: for all z with z ρ < r ext : h0) hz) M h r ext ρ h0) ; 6) for all z with z ρ < r ext : hz) h0) M h r ext +ρ h0). 7) Assume w.l.o.g. that for f we have n 0 f) < n f) otherwise consider f instead). Let the zeros z, z 2,..., z n0 of f K 2 lying in z rf) be given by z j = γ j re iϑ 0, 0 γ j, ϑ j [0, 2π], r := rf), and consider the product of fz) with Blaschke factors B j z) chosen as B j z) := r rγj e iϑj z. z γ j re iϑ j Each Blaschke factor is unimodular on z = r, and of modulus at most r r cr cr r for z = cr, 0 < r < c <, 8) as a discussion of the parameter γ j shows. Multiply the function fz) by Bz) := n 0 j= B jz) to cancel all roots inside the unit disc. The analytic function fz)bz) is zero-free, hence we may apply the upper estimate 7) to hz) := /fz)bz)) to estimate its minimum modulus. First, from 7) we obtain the inequality r ext ρ fz)bz) r ext +ρ f0)b0) max Therefore, taking reciprocals, fz)bz) f0)b0) r ext ρ r ext +ρ w =r ext r ext +ρ fw)bw), where z = ρ < rext. min fw)bw) r ext +ρ, where z = ρ < rext. w =r ext Since Bz) is unimodular on z = r ext = r the following inequality holds for an arbitrary z of modulus ρ: min fz) f z) rext+ρ rext+ρ max Bz) z =ρ f0)b0) r ext ρ This implies, using the estimate 8) for the n 0 factors of the Blaschke product Bz), and the relation /B0) r) n 0, that 5.

6 min fz) f z) rext+ρ r r ρ ρ r ) n0 r ext +ρ r) n 0 f0) r ext ρ. 9) Put r := , hence r ext := r Choose an intermediate value as ρ = By 4), there exists z on the circle z = ρ such that f z). We may estimate the minimum modulus of f on z = r = r ext distinguishing the following three cases..) Assume f0) > Then, from 9) with f z) ) we get min fz) ) n0 r ext +ρ r r ρ r) n r ext ρ 0 ρ r f0) ) ) ) , since the upper bound decreases with growing n 0. 2.) If f0) < , consider f with modulus at the origin at least , and n 2 zeros. Using the inequality 9) with f in place of f for some z with f z) ), and n in place of n 0, we obtain min fz) ) n r ext +ρ r r ρ r) n r ext ρ ρ r f0) ) ) ) , since the upper bound decreases with growing n 2. 3.) Consider finally the case f0) The function F z) := fz) fz)) has at least three zeros, while F 0) ). By 4), it exists a point z on the circle of radius ρ = such that F z) /4. Using this information in 9) with F in place of f hence n 0 = n 0 F ) 3) we obtain ) r ext +ρ n0 +n min F z) /4 rext+ρ r r ρ r) n 0+n max z =ρ ρ r F 0) ) ) ) ) , since the upper bound decreases with growing n 0 + n 3. 6 r ext ρ

7 This estimate of F z) = fz) fz)) implies that at least one of the functions f and f is at one point of the circle z = r ext no larger in modulus than The three cases considered above allow us to conclude that for at least one f {f, f} we have min fz) < We may assume that g ) = fκ )) has modulus at most at the origin, or else consider a suitable rotation of κ ). This leads to the maximum modulus estimate of f {f, f} on r < z < r via 2) and 3) with b := e π 6) max r< z < r fz) 6 exp π 2 π2 6 exp ln + b) lnr) )) b =: M. 0) With this bound for fz) over r < z < r, i.e. a bound for one of the functions f and f, we obtain a lower bound for the modulus of f on a new circle z = ρ where ρ r, r) will be specified later) via the following estimate: inequality 6) for hz) := fz)bz), r ext = r yields for z with z = ρ that fz) max z =ρ r+ρ Bz) ) f0)b0) r ρ ) r ρ max Bw) fw). w =r ext As Bw) is unimodular for w = r ext = r, we use 0) to obtain fz) max z =ρ r+ρ Bz) ) f0)b0) r ρ M + ) r ρ. ) With n 0 and f0) , the Blaschke factor estimate 8) in ) yields for all z with z = ρ the lower bound fz) ) n0 ρ r r r ρ [ ] r+ρ ) n r ρ 0 M + ) r ρ. r Choose the intermediate radius as ρ := to obtain for r = on z = ρ the estimate fz) ) n0 ρ r r r ρ f0) r+ρ ) n r ρ 0 r M + ) r ρ / )

8 Similarly f0) 0.05 and n f) = n 0 f) 2 yield for all z on z = ρ the lower estimate fz) ) n ) ρ r r+ρ n r r r ρ f0)) r ρ M + ) r ρ ) 2 ρ r ) r+ρ 2 r ρ f0)) M + ) r ρ r r ρ r / ) This contradicts the fact that the functions f and f simultaneously intersect the interval 0, ) on z = ρ, i.e. contradicts equation 5). This establishes the following result. Theorem 3 A 2 > Acknowledgement. The author wishes to thank Editor Stephan Ruscheweyh for several hints and remarks which helped to improve the presentation of results. References [] Batra, P. On Small Circles Containing Zeros and Ones of Analytic Functions. Complex Variables: Theory and Applications, 49):787 79, [2] Bieberbach, L. Über die Verteilung der Eins- und Nullstellen analytischer Funktionen. Mathematische Annalen, 85:4 48, 922. [3] Blondel, V.D.; Rupp, R.; Shapiro, H.S. On Zero and One Points of Analytic Functions. Complex Variables, 28:89 92, 995. [4] Blondel, V.; Sontag, E.; Vidyasagar, M.; Willems, J.C., Open Problems in Mathematical Systems and Control Theory, Springer, London, 999. [5] Carathéodory, C. Sur quelques généralisations du théorème de M. Picard. Comptes Rendues des Seances de l Académie des Sciences, 4:23 25, 904. [6] Gol dberg, A.A. On a theorem of Landau type. Teor. Funkts., Funkts. Anal. Prilozh., 7: , 973. [7] Hayman, W. K. Some remarks on Schottky s theorem. Proc. Cambridge Phil. Soc., 43: ,

9 [8] Hayman, W. K. Subharmonic Functions, vol. 2. Academic Press, London, 989. [9] Hempel, J. A. Precise Bounds in the Theorems of Schottky and Picard. J. Lond. Math. Soc., II. Ser., 2: , 980. [0] Jenkins, J.A. On a problem of A. A. Gol dberg. Ann. Univ. Mariae Curie- Sk lodowska, Sect. A, pp , 983. [] Landau, E. Über eine Verallgemeinerung des Picardschen Satzes. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, pp. 8 33, 904. [2] Ostrowski, A.M. Asymptotische Abschätzung des absoluten Betrages einer Funktion, die die Werte 0 und nicht annimmt. Comment. Math. Helvet., 5:55 87, 933. [3] Pólya, G.; Szegö, G. Aufgaben und Lehrsätze aus der Analysis I. Springer- Verlag, Berlin, 970. [4] Rupp, R. A Covering Theorem for a Composite Class of Analytic Functions. Complex Variables: Theory and Applications, 25:35 4, 994. [5] Schottky, F. Über den Picard schen Satz und die Borel schen Ungleichungen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, Berlin, pp , 904. [6] Zhang, S. On Explicit Bounds in Schottky s Theorem. Complex Variables: Theory and Applications, 3:6 7, 990. Prashant Batra batra@tuhh.de ADDRESS: Institute for Computer Technology, Hamburg University of Technology, D-207 Hamburg, Germany. 9