The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents

Depends on

• Real powers for positive bases, with the zero-base positive-exponent convention
• Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm
• The exponential addition formula $\exp(x+y)=\exp(x)\exp(y)$
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$

Used by

• Compatible extensions from the finite simple core Corollary
• Gautschi's inequality for the real Gamma function Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• Lyapunov central limit theorem Corollary
• Polynomial and exponential minimax estimation rates Corollary
• The polynomial Rödl property implies the Erdős–Hajnal property Corollary
• The volume of a radius-r closed n-ball is π^n/2rⁿ/Γ(n/2+1) Corollary
• Finite thresholds rule out an r-independent VC covering bound Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Covering, packing, bracketing numbers, and metric entropy Definition
• Integral geometric layers of a decreasing block partition Definition
• Quasisymmetric homeomorphisms of the line and circle Definition
• A Dirac mass has precisely sufficiently negative Sobolev order Example
• Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example
• Coordinate vectors converge weakly to zero in ell p Example
• Dilation determines the Riesz-potential target exponent Example
• Exponential singular decay, logarithmic balls, and spectral gaps Example
• Exponential singular-value decay and an exponentially weighted truth class Example
• Newton shell theorem from harmonic mean values Example
• Newtonian potential of radial compact data Example
• Nonidentical strong law under summable normalized variances Example
• Polynomial singular decay with power and logarithmic source balls Example
• Polynomial singular-value decay and a power Gaussian prior Example
• Power maps, endpoint distortion, and a non-Möbius quasisymmetric circle map Example
• Sharp Sobolev threshold for a radial power Example
• The Koch snowflake is a non-rectifiable quasicircle Example
• Tightness from a uniform moment bound Example
• A bounded-domain Dirichlet Green function is unique and positive Lemma
• A large Y-part in a structural comb partition yields the clique-or-stable-set outcome Lemma
• A wide integral geometric layer forces the complete-or-anticomplete property-(*) blockade Lemma
• Admissible parameters for the density recursion Lemma
• Backward-Euler fractional weights and positive partial sums Lemma
• Clarkson inequalities in both exponent ranges Lemma
• Elementary lower and upper bounds on a unit cube Lemma
• Finite geometric error bound from a one-step recurrence Lemma
• Generating-coefficient estimate for the quadratic CQ datum Lemma
• Increasing the exponent past finite measure gives zero Lemma
• Integral geometric layers exist, cover the partition, and retain the required cutoff bounds Lemma
• Local special copy trichotomy Lemma
• Log-convex solutions of the Gamma recurrence obey the Bohr--Mollerup factorial squeeze Lemma

…and 50 more results.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)