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
• 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
• Integral geometric layers of a decreasing block partition Definition
• Coordinate vectors converge weakly to zero in ell p Example
• Nonidentical strong law under summable normalized variances Example
• Tightness from a uniform moment bound Example
• 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
• Clarkson inequalities in both exponent ranges Lemma
• Elementary lower and upper bounds on a unit cube 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
• Qid maximal blowup trichotomy Lemma
• Special copy trichotomy produces a restricted blockade Lemma
• Successive small integral geometric layers contradict a large X-part Lemma
• The p-functional need not be a norm for 0 < p < 1 Proposition
• A tau-critical graph has no wide pure blockade with cograph pattern Theorem
• Alon–Pach–Solymosi: if H₁ and H₂ have the Erdős–Hajnal property, so does the graph obtained from H₁ by substituting H₂ for a vertex Theorem
• An entire function of polynomial growth is a polynomial Theorem
• An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem
• Change of base and inversion of the positive-base real exponential Theorem
• Every finite family with the Erdős–Hajnal property is viral Theorem
• Finite variance logarithmic rate for iid sums Theorem
• For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k Theorem
• Holder's inequality for finite sums and conjugate real exponents Theorem
• ℓᵖ includes into ℓʳ for p < r Theorem
• Minkowski's inequality for finite sums and real exponent p greater than one Theorem
• Quantitative density theorem for ell divisive graphs Theorem
• Standard Maclaurin expansions Theorem
• Strong law under summable normalized variances Theorem
• The C5-free graphs satisfy a polynomial kappa bound Theorem
• The Cantor set has dimension log 2 / log 3 and critical measure one Theorem
• The Erdos-Hajnal property is equivalent to the large-cograph, large-perfect, and kappa formulations Theorem

…and 8 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)