Search the corpus

    JHU analysis exams (compiled packet)

    SRC-JHU-ANALYSIS-EXAMS

    Area
    Complex Analysis, Real Analysis
    Status
    draft
    Source
    Contents

    299 problems.

    Qualifying Exam in Analysis, September 4, 2019, 9am–12noon (pp. 1–2)

    1. Convergence of positive continuous functions and Fatou-type inequality
    2. \(L^1\) convergence implies existence of a.e. convergent subsequence
    3. The convolution of two \(L^2(\mathbb R)\) functions is continuous
    4. Every closed convex subset of a Hilbert space has a unique element of minimal norm
    5. \(\int_0^\infty\frac{\log x}{x^2+2}\,dx\) by contour integration
    6. Polynomial growth of entire functions and derivatives of normal families
    7. A holomorphic function on the punctured disk dominated by a power of the logarithm
    8. Nonempty closed convex sets in a Hilbert space have unique minimum-norm points

    ANALYSIS QUALIFYING EXAM: SPRING 2019 (p. 3)

    1. Weak convergence plus pointwise decay need not imply strong convergence in \(L^2(\mathbb R^d)\)
    2. Separate measurability and section functions of a measurable function of two variables
    3. Monotone convergence of continuous functions is uniform iff limit is continuous
    4. The sine integral converges, but \(\sin x/x\notin L^1(1,\infty)\)
    5. Automorphisms of the upper half plane
    6. Holomorphic off the real line and continuous everywhere implies entire
    7. Evaluating \(\int_0^\infty x^{-1/3}/(1+x)\,dx\)
    8. Punctured unit disk and annulus are not conformally equivalent

    Johns Hopkins University, Department of Mathematics, Real and Complex analysis (undated, pp. 4–5)

    1. Dyadic averages of an \(L^1\) function converge almost everywhere
    2. Unit ball of \(L^2\) is not compact
    3. Banach space characterization via absolutely convergent series
    4. Entire functions with pointwise modulus bound are proportional
    5. Evaluating the sinc integral by contour integration
    6. Maximum modulus property forces holomorphic functions to be constant
    7. Bergman space is a Hilbert space

    Johns Hopkins University, Department of Mathematics, Real and Complex analysis (undated, pp. 6–7)

    1. Dyadic averages of an \(L^1\) function converge almost everywhere
    2. Weak convergence plus norm convergence implies strong convergence in \(L^2\)
    3. Holder inequality, Young inequality, and \(L^p\) Banach spaces
    4. Entire function with zero coefficient at every center is polynomial
    5. Meromorphic function with pole has no holomorphic logarithm
    6. Holomorphic function in annulus approximable by polynomials extends to disc
    7. Bergman space embedding, normal family, and removable singularity

    Qualifying Exam - Analysis-Fall 2017 (p. 8)

    1. Eventual equicontinuity at a continuity point of a monotone limit
    2. Continuity of translations in \(L^p(\mathbb R^n)\)
    3. Small sets for a Radon measure via the Vitali covering lemma
    4. The Schwarz lemma with a zero at an arbitrary point of the disk
    5. Entire functions with polynomially bounded real part are polynomials
    6. The two-circles bound for subharmonic functions

    Qualifying Exam - Analysis - Spring 2017 (p. 9)

    1. Layer-cake formula with an absolutely continuous weight: \(\int_{\RR}\varphi\circ f\,dx=\int_0^\infty m(\{f>t\})\varphi'(t)\,dt\)
    2. Weak and strong convergence in Hilbert space, with Banach-Saks subsequence means
    3. Borel-Cantelli: measurable sets of summable measure
    4. Extremal second derivative among holomorphic maps into the unit disk
    5. Holomorphic functions on the punctured plane with a two-sided growth bound
    6. Unit modulus on the boundary circle forces the image to contain the disk

    Qualifying Exam - Analysis - Fall 2016 (p. 10)

    1. \(L^2\) convergence has an almost-everywhere convergent subsequence
    2. Translates of a positive-measure set cover almost all of the line
    3. Norm-one sequences in an infinite-dimensional Hilbert space: subsequences
    4. Uniform limits on compact subsets of harmonic functions are harmonic
    5. A univalent branch of \(\sqrt{f(z^2)}\)
    6. Zero-count stability when the limit has no boundary zeros

    QUALIFYING EXAM SPRING 2016 - ANALYSIS (p. 11)

    1. Absolute continuity of the Lebesgue integral
    2. Hardy-Littlewood maximal function bound via Vitali covering
    3. Layer-cake formula with a \(C^1\) weight on \([0,1]\)
    4. Bergman space is a Hilbert space
    5. Holomorphic self-map of disk with two fixed points is identity
    6. Entire function with unbounded zero set has essential singularity at infinity
    7. Zeros of \(2z^5-6z^2+z+1\) in a closed annulus

    QUALIFYING EXAM - ANALYSIS - FALL 2015 (p. 12)

    1. A sequence converging to \(0\) in \(L^2\) has a subsequence converging almost everywhere
    2. Monotone and absolutely continuous functions: two true-or-false statements
    3. Products of orthonormal bases span the product \(L^2\) space
    4. Uniform approximation on the boundary preserves the zero count
    5. Entire functions with nonpositive real part are constant
    6. \(f^2\) and \(f^3\) holomorphic imply \(f\) holomorphic
    7. The logarithmic integral \(\int_0^\infty (\log x)^2/(1+x^2)\,dx\)

    Qualifying Exam - Analysis - Spring 2015 (p. 13)

    1. A.e. convergence plus convergence of \(L^2\) norms implies \(L^2\) convergence
    2. Mollification does not increase the \(L^2\) norm
    3. An increasing convex function on \([0,1]\) is the integral of its derivative
    4. \(fg\in L^1\) for all \(g\in L^2\) implies \(f\in L^2\)
    5. Componentwise rigidity of a holomorphic conjugate product
    6. Nonconstant entire functions have dense image
    7. All zeros of \(z^5+3z^3+7\) lie in \(|z|<2\)
    8. Logarithmic bound on holomorphic function in annulus

    Qualifying Exam - Analysis - Fall 2014 (p. 14)

    1. Convergence of \(L^1\) norms upgrades a.e. convergence to \(L^1\) convergence — Problem 1
    2. The Hardy-Littlewood maximal function is weak (1,1) — Problem 2
    3. Compact operators are norm limits of finite-rank operators — Problem 3
    4. Uniform convergence on the boundary gives uniform convergence on the region — Problem 4
    5. Evaluating \(\int_0^\infty\cos(ax)/(1+x^2)^2\,dx\) — Problem 5
    6. Entire functions bounded by a nonvanishing entire function are constant multiples — Problem 6
    7. Automorphisms of the unit disk are linear fractional transformations — Problem 7

    QUALIFYING EXAM - ANALYSIS - SPRING 2014 (p. 15)

    1. Egoroff’s theorem for a dominated sequence, proved without invoking Egoroff — Problem 1
    2. Integral equals integral of distribution function — Problem 2
    3. Lipschitz condition equivalent to absolute continuity with bounded derivative — Problem 3
    4. Multiplication operator on \(L^2\) is self-adjoint but not compact — Problem 4
    5. Schwarz-Pick type inequality for holomorphic self-map of disk — Problem 5
    6. Fourier transform of the sinc function — Problem 6
    7. Zero derivative implies not one-to-one in any neighborhood — Problem 7
    8. An entire function of polynomial growth is a polynomial — Problem 8

    QUALIFYING EXAM - ANALYSIS - FALL 2013 (p. 16)

    1. \(L^1\) integrability of integral transform — Problem 1
    2. Measurable set with dense complement and no isolated points — Problem 2
    3. Smooth compactly supported functions are dense in \(L^2\) — Problem 3
    4. Nonzero smooth compactly supported function with compactly supported Fourier transform — Problem 4
    5. \(\int_0^\infty\frac{\cos u}{u^4+1}\,du\) — Problem 5
    6. Distinct solutions of \(e^{z^2}=5z^5\) in the closed unit disk — Problem 6
    7. Entire functions bounded by \(|z|^2|\operatorname{Im}z|^2\) at infinity — Problem 7

    Qualifying Exam - Analysis, May, 2013 (pp. 17–18)

    1. A continuous \(f\) with \(f^2\) holomorphic is holomorphic
    2. Zeros of \(z^4-6z+3\) in the unit disk and an annulus
    3. Growth bound on the disk from an upper bound on the real part
    4. Evaluating \(\int_0^\infty x^{1/2}/(4+x^2)\,dx\)
    5. Sum of two positive measure sets contains an interval
    6. Riemann-Lebesgue lemma and convolution identity
    7. \(\|f_n\|_2\to0\) implies an a.e.-convergent subsequence
    8. Weak convergence plus norm convergence implies strong convergence in \(L^2\)

    ANALYSIS QUALIFYING EXAM, FALL 2012 (pp. 19–20)

    1. Zeros of a polynomial in the annulus from one to three
    2. Residue computation of the Fourier transform of \(\frac{1}{x^{2}+1}\)
    3. Maximal derivative of a disk self-map at an interior point
    4. Continuity across a segment plus holomorphy off it implies entire
    5. \(L^1\) convergence gives an a.e.-convergent subsequence but not uniform convergence
    6. Fatou’s lemma, the dominated convergence theorem, and a counterexample
    7. Convolutions of \(L^2\) functions and convolution operators on \(L^1\)
    8. \(L^\infty\) and weak-\(L^1\) bounds imply \(L^2\) boundedness

    Analysis Qualifying Exam, May 9, 2012 (p. 21)

    1. Residue evaluation of \(\int_0^\infty (1+x^2)^{-2}\,dx\)
    2. Unique fixed point of a disk map whose boundary values lie in the open disk
    3. Removable singularity when real part is bounded below
    4. Half-plane preservation and two interpolation values force a translation
    5. Weak and strong convergence of squares in \(L^2\)
    6. Absolutely convergent series of \(L^1\) functions converges a.e. and in \(L^1\)
    7. Integral transform is absolutely continuous and vanishes at infinity
    8. Fourier transform is continuous and vanishes at infinity

    ANALYSIS QUALIFYING EXAM, SEPTEMBER 2011 (pp. 22–23)

    1. \(\int_\gamma \frac{dz}{z^3\cos z}\) for \(|z-1|=2\)
    2. A harmonic function on the plane bounded below is constant
    3. Finite valence excludes an essential singularity
    4. Unique normalized rotations of a simply connected bounded domain
    5. An \(L^2\)-bounded sequence in \(L^2\) of the line without convergent subsequences
    6. Fourier-transform integrals over finite- and infinite-measure sets
    7. Small-level decay of the distribution function of an \(L^p\) function
    8. No convolution identity in \(L^1(\RR^n)\)

    ANALYSIS QUALIFYING EXAM, MAY 2011 (pp. 24–25)

    1. Entire functions with unit modulus on the unit circle are monomials
    2. Weighted logarithmic derivative around two prescribed zeros
    3. A holomorphic self-map of a bounded domain has derivative at most one at a fixed point
    4. A logarithmic upper bound forces an entire harmonic function to be constant
    5. The running maximum of a bounded C1 sequence converges uniformly
    6. Absolutely summable Fourier coefficients give a continuous representative
    7. \(L^p\) norms converge to the \(L^\infty\) norm on a probability space
    8. Weak \(L^2\) convergence: lower semicontinuity and strong convergence from norm convergence

    Analysis Qualifying Exam, Fall 2010 (pp. 26–27)

    1. \(L^2\) convergence with Gaussian weight
    2. Characteristic-function convolution and one-sided differentiation averages
    3. Schur’s test for integral operators
    4. Translation operator on \(L^2\) has no real eigenvalues
    5. Residues on a twice-traversed limaçon
    6. Vanishing first jets at the points \(1/n\)
    7. Entire functions dominated by \(|\operatorname{Re}z|^2+|z|^{3/2}\)
    8. The quadratic Schwarz bound for a two-sheeted branched disk map

    ANALYSIS QUALIFYING EXAM, MAY 2010 (pp. 28–29)

    1. Removable singularity under the bound \(|f(z)|\le|z|^{-1/2}\)
    2. All integral values over simple closed curves avoiding two poles
    3. The Bergman space of the unit disk
    4. Harmonic conjugate on the exterior of the unit disk up to a logarithmic term
    5. Uniform \(H^\sigma\) bounds pass to an \(L^2\) limit
    6. Convolution with a compactly supported kernel maps \(L^p\) to \(L^q\) exactly when \(p\le q\)
    7. Differentiation under the integral sign with a dominated partial derivative
    8. Continuity of overlap measure and Steinhaus’ theorem

    ANALYSIS QUALIFYING EXAM, SEPTEMBER 2009 (pp. 30–31)

    1. Pointwise versus uniform convergence of continuous functions
    2. Bilinear forms with polynomially weighted coefficients on l2
    3. Dyadic averages of an \(L^1\) function converge almost everywhere
    4. Integral of the derivative of a monotone function is bounded by its total increase
    5. Complex differentiability at a point without local holomorphy
    6. Removability and the Taylor radius of \(z\cot z\)
    7. The sharp derivative bound for an upper-half-plane map with \(f(i)=1/2\)
    8. Harmonic conjugate on the punctured disk up to a logarithmic term

    ANALYSIS QUALIFYING EXAM, MAY 2009 (pp. 32–33)

    1. Meromorphic functions with a logarithmic bound divided by \(|z|\)
    2. The unique right-half-plane solution of \(z+e^{-z}=2+i\)
    3. Uniformly convergent holomorphic sequences with a single zero each
    4. The Fourier transform of \(1/(1+x^2)\) at frequency one
    5. Convolution of two \(L^2\) functions is bounded and continuous
    6. Uniform convergence for an approximation to the identity on the circle
    7. \(L^p\) convergence implies convergence in measure on the unit interval
    8. Successive moment ratios converge to the essential supremum

    PROBLEMS FOR ANALYSIS QUALIFYING EXAM, Fall 2008 (p. 34)

    1. Iteration of s -> s^2 on [0,1]: pushforward measures and their weak limit
    2. Integral operators on \(L^p\) via Schur-type bounds
    3. Separability of the Banach space l-infinity
    4. \(\int_0^\infty\frac{\log x}{(x^2+4)^2}\,dx\) by residues
    5. Counting solutions of e^z = 3z^7 in the unit disk
    6. Entire functions of polynomial growth are polynomials
    7. Holomorphic bijections onto a simply connected domain agreeing at 0

    PROBLEMS FOR ANALYSIS QUALIFYING EXAM, SPRING 2008 (p. 35)

    1. Convolutions of characteristic functions of finite-measure sets
    2. \(\|f\|_p\to\|f\|_\infty\) on \([0,1]\), with a counterexample on \(\RR\)
    3. Absolute convergence of the Fourier series of a C1 periodic function
    4. Boundedness of an operator on l2 from its matrix coefficients
    5. Entire functions bounded by a nonvanishing entire function are constant multiples
    6. Conformal self-maps of the disk agreeing at two points coincide
    7. A holomorphic function with a double zero takes nearby values at least twice
    8. Derivative bounds and extremal sequences for families of disk-valued holomorphic maps

    ANALYSIS QUALIFYING EXAM, FALL 2007 (p. 36)

    1. Complex analyticity of a real polynomial via the Cauchy-Riemann equations
    2. Entire functions bounded by the exponential
    3. Unit boundary modulus forces at least two annular zeros with multiplicity
    4. The integral of \(1/(1+x^n)\) by residue calculus
    5. Membership examples separating \(L^1\) and \(L^2\)
    6. An open dense subset of R with measure one
    7. A summably integrable sequence converges to zero almost everywhere
    8. Compactly supported \(L^1\) convolution need not be uniformly continuous

    PROBLEMS FOR ANALYSIS QUALIFYING EXAM, SPRING 2007 (p. 37)

    1. Zeros of a sextic in the unit disk
    2. The Poisson kernel integral by residues
    3. Iterates of a non-rotational disk map vanish at the fixed point
    4. A surjective holomorphic map from the disk onto \(\mathbb C\)
    5. Membership of \(1/x\) in \(L^p(0,\infty)\)
    6. \(L^4\) functions with vanishing \(L^1\) norm converge weakly
    7. Weakly convergent \(L^2\) sequences with unbounded norms
    8. A function and its Fourier transform cannot both have compact support

    ANALYSIS QUALIFYING EXAM, SEPTEMBER 2006 (pp. 38–39)

    1. The integral of \(1/(x^4+4)\) by residues
    2. Hurwitz theorem for uniformly convergent holomorphic sequences with one zero
    3. Holomorphic maps into the upper half plane and bounded harmonic functions on a slit plane
    4. Maximal derivative of a disk map into the upper half plane
    5. Finite-dimensionality of trigonometric-polynomial subspaces of C[0, pi]
    6. Oscillatory integral is o(1/t) under endpoint vanishing
    7. \(L^2\) boundedness and vanishing \(L^1\) norm imply weak convergence to \(0\)
    8. Unique norming point for a functional on a space satisfying the parallelogram inequality

    ANALYSIS QUALIFYING EXAM, MAY 2006 (pp. 40–41)

    1. No holomorphic logarithm of a meromorphic function with a pole
    2. Zeros of a septic in the disks of radius one and two
    3. Entire functions bounded by \(|z|^{3/2}\) outside the unit disk
    4. Monotone harmonic functions converging to the constant one
    5. Derivative range of a half-plane self-map at a fixed point
    6. Poles and residues of \(e^{\pi z}/(z^2+1)^2\)
    7. Separating \(L^1\) and \(L^2\) on \(\RR\) and \((0,1)\)
    8. \(L^1\cap L^2\subseteq L^p\) for \(1\le p\le2\)
    9. The Fourier transform of an \(L^1\) function is uniformly continuous and vanishes at infinity
    10. Translation continuity of \(L^1\) functions on the line
    11. Boundedness of a weakly convergent sequence in \(L^2([0,1])\)
    12. Cesaro means of an orthonormal sequence converge to zero a.e.

    ANALYSIS QUALIFYING EXAM, SEPTEMBER 2005 (pp. 42–43)

    1. \(\|f_n\|_2^2\le n^{-2}\) forces \(f_n\to0\) a.e.
    2. Translation continuity of \(L^1\) functions on the line
    3. Weak convergence to zero on the circle
    4. Weak lower semicontinuity of the norm and strong convergence
    5. The keyhole integral of \(1/(x^{1/3}(1+x))\)
    6. Automorphisms of the right half-plane and of \(\mathbb C\)
    7. Zeros of \(z^9+z^5-8z^3-z+2\) between two circles
    8. Surjective holomorphic maps between the right half-plane and the plane

    ANALYSIS QUALIFYING EXAM SPRING 2005 (p. 44)

    1. Smoothness and decay of the Fourier transform of e^{-|x|}
    2. An integral operator on L^1[0,1] with kernel 1/t
    3. \(L^p\) inclusions on finite- and infinite-measure spaces
    4. Weak and strong convergence of an orthonormal basis and its Cesàro means
    5. Nonzero C_c^∞ function with compactly supported Fourier transform
    6. Residue evaluation of a rational-times-sine integral
    7. Holomorphic bijection from the quadrant to the unit disk
    8. Zeros equal poles for a meromorphic function with positive real boundary values

    Undated analysis exam, closed book, two and a half hours, seven problems (pp. 45–47)

    1. \(C([0,1])\) is a Banach space; \(\sin nx\) converges weakly to \(0\) in \(L^2\) but not strongly
    2. Countable additivity of the integral; densities as signed Borel measures; a computed mass
    3. Direct proof that the Fourier transform of an \(L^1\) function is uniformly continuous and vanishes at infinity
    4. Integrability of \(\cos x/(1+x^2)\) and evaluation of \(\int_0^\infty\cos x/(1+x^2)\,dx\)
    5. Nonreal solutions of \(z^3+z^4=2\) in \(|z|<2\)
    6. Entire functions bounded by \(3\log|z|\) at infinity
    7. The integral of \(e^z/z^3\) around \(|z-1|=2\)

    Undated analysis exam, seven problems, instructions citing the syllabus (pp. 45–47)

    1. \(g_n(x)=\sin^2(nx)\): pointwise, strong \(L^1\), and weak \(L^1\) convergence
    2. Maximum of the exponential of the mean log under a mean-one constraint
    3. Decay, square-integrability, continuity, and boundedness of the Fourier transform of \((1+\xi^2)^{-\alpha}\)
    4. Residue evaluation of \(f_1(x)=\int_{\mathbb R}(1+\xi^2)^{-1}e^{2\pi i x\xi}\,d\xi\)
    5. Contour integrals of \(1/f\) at simple and double zeros
    6. A boundary contraction of the disk has exactly one interior fixed point
    7. A nonidentity disk automorphism fixing \(1/3\)

    Undated analysis exam, two hours, six equally weighted questions (pp. 45–47)

    1. Absolute continuity of the Lebesgue integral: small-measure sets have small integrals
    2. Wirtinger’s inequality on \([0,\pi]\) for functions vanishing at the endpoints
    3. The operator \(Tf(x)=x^{-1/p}\int_0^xf\) maps \(L^q\) boundedly into \(C^0\)
    4. Zeros of \(2z^5+8z-1\) in \(1<|z|<2\)
    5. Bergman-type area integral of the derivative on a punctured disk: bounded vs unbounded case
    6. A positive harmonic function on the punctured plane is constant

    SPRING 2003 COMPLEX ANALYSIS QUALIFYING EXAM (p. 48)

    1. Meromorphic functions on the sphere and rational boundary-modulus extensions
    2. A second zero of a disk self-map satisfies \(|w|\geq|f'(0)|\)
    3. Recovering zeros from logarithmic-derivative moments
    4. Rouché’s theorem, segment estimates, and a univalence criterion

    Real Analysis Qualifying Exam, Fall 2002 (p. 49)

    1. A continuous nowhere differentiable function built from the triangle wave
    2. Monotone convergence for an increasing sequence with bounded integrals
    3. Equicontinuity and the Arzela-Ascoli theorem on a Sobolev family
    4. Unique nearest-point projection onto a closed convex subset of a Hilbert space
    5. The sum of the odd sine series and the Basel problem

    Complex Analysis Core Qualifying Exam, Fall 2002 (p. 50)

    1. Failure of holomorphic and meromorphic extension from an annulus
    2. Holomorphic and smooth equivalence of an annulus with a punctured disk
    3. Entire functions bounded by the square of the modulus
    4. A disk automorphism sending \(1/2\) to \(-1/\pi\)
    5. Simplicity and small-disk zero count for \(z^7+z^3+1/16\)
    6. Evaluating \(\int_0^\infty u^2/(u^6+1)\,du\)

    Real Analysis Qualifying Exam (p. 51)

    1. Quantitative Riemann-Lebesgue decay under vanishing endpoint conditions
    2. A pointwise-convergent subsequence of sine sums with continuous limit
    3. The \(1/5\)-Hölder norm \(|f(0)|+[f]_{1/5}\) is complete on \([0,1]\)
    4. A function whose every set-integral is bounded by measure is at most 1 a.e.
    5. \(\|f\|_p\to\|f\|_\infty\) as \(p\to\infty\) for \(f\in L^1\cap L^\infty\)
    6. The principal value distribution is the derivative of log |x|

    Complex Analysis Core Qualifying Exam, Spring 2002 (p. 52)

    1. Entire functions whose image omits a ray
    2. Evaluating \(\int_0^{2\pi}(a^2+\cos^2x)^{-1}\,dx\)
    3. Classification of simply connected regions in the Riemann sphere
    4. Even-part Schwarz bound for a normalized disk self-map
    5. Negative Laurent coefficients of \(1/\sin z\) on \(\pi<|z|<2\pi\)
    6. Mobius transformations carry lines and circles to lines and circles

    Real Analysis Qualifying Exam, Fall 2001 (pp. 53–54)

    1. Uniform continuity of continuous functions with limit at infinity
    2. Convolution of functions vanishing at infinity
    3. \(L^{4/3}\) boundedness plus convergence in measure implies \(L^1\) convergence
    4. Step function approximation converges in \(L^1\)
    5. \(L^p\) inclusion relationships

    COMPLEX ANALYSIS CORE QUALIFYING EXAM, FALL 2001 (p. 55)

    1. Meromorphic functions on the Riemann sphere are rational
    2. The unique right-half-plane solution of \(z-\alpha=e^{-z}\) is real
    3. Evaluating \(\int_0^\infty(1+x^3)^{-1}\,dx\)
    4. Schwarz-Pick: holomorphic self-map of disk maps smaller disks into smaller disks
    5. Growth bound for a disk map into a vertical strip

    REAL ANALYSIS QUALIFYING EXAM, SPRING 2001 (p. 56)

    1. Mollifier convolutions converge in \(L^p\) but not in \(L^\infty\)
    2. Absolute continuity of the Lebesgue integral
    3. Lower semicontinuous functions and lower Riemann sums
    4. Interpolation inequality between the \(L^{4/3}\), \(L^2\), and \(L^4\) norms
    5. Integral operators with continuous kernel map the unit ball of C[0,1] to a precompact set

    COMPLEX ANALYSIS CORE QUALIFYING EXAM, SPRING 2001 (p. 57)

    1. An entire function dominated by another is its scalar multiple
    2. Zeros of \(2z^5+8z-1\) in \(1<|z|<2\)
    3. Convergence on a dense subset implies pointwise convergence for bounded holomorphic functions
    4. A positive harmonic function on the punctured plane is constant
    5. A nonconstant annulus map of constant boundary modulus has at least two zeros
    6. Gauss–Lucas theorem for polynomial critical points

    Browse these problems in source order