mathematics

Logarithmically convex function

Logarithmically convex function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Logarithmically convex function rather than just read about it. In short: In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function. Definition Let X be a convex subset of a real vector space, and let f : X → R be a function taking non-negative values.

Key takeaways

Logarithmically convex function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Logarithmically convex function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Logarithmically convex function from memory before moving on to harder problems.

Reference excerpt

In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function.

Definition Let X be a convex subset of a real vector space, and let f : X → R be a function taking non-negative values. Then f is:

Logarithmically convex if log ∘ f {\displaystyle {\log }\circ f} is convex, and Strictly logarithmically convex if log ∘ f {\displaystyle {\log }\circ f} is strictly convex. Here we interpret log ⁡ 0 {\displaystyle \log 0} as − ∞ {\displaystyle -\infty } . Explicitly, f is logarithmically convex if and only if, for all x1, x2 ∈ X and all t ∈ [0, 1], the two following equivalent conditions hold:

log ⁡ f ( t x 1 + ( 1 − t ) x 2 ) ≤ t log ⁡ f ( x 1 ) + ( 1 − t ) log ⁡ f ( x 2 ) , f ( t x 1 + ( 1 − t ) x 2 ) ≤ f ( x 1 ) t f ( x 2 ) 1 − t . {\displaystyle {\begin{aligned}\log f(tx_{1}+(1-t)x_{2})&\leq t\log f(x_{1})+(1-t)\log f(x_{2}),\\f(tx_{1}+(1-t)x_{2})&\leq f(x_{1})^{t}f(x_{2})^{1-t}.\end{aligned}}}

Similarly, f is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all t ∈ (0, 1). The above definition permits f to be zero, but if f is logarithmically convex and vanishes anywhere in X, then it vanishes everywhere in the interior of X.

Equivalent conditions If f is a differentiable function defined on an interval I ⊆ R, then f is logarithmically convex if and only if the following condition holds for all x and y in I:

log ⁡ f ( x ) ≥ log ⁡ f ( y ) + f ′ ( y ) f ( y ) ( x − y ) . {\displaystyle \log f(x)\geq \log f(y)+{\frac {f'(y)}{f(y)}}(x-y).}

This is equivalent to the condition that, whenever x and y are in I and x > y,

( f ( x ) f ( y ) ) 1 x − y ≥ exp ⁡ ( f ′ ( y ) f ( y ) ) . {\displaystyle \left({\frac {f(x)}{f(y)}}\right)^{\frac {1}{x-y}}\geq \exp \left({\frac {f'(y)}{f(y)}}\right).}

Moreover, f is strictly logarithmically convex if and only if these inequalities are always strict. If f is twice differentiable, then it is logarithmically convex if and only if, for all x in I,

f ″ ( x ) f ( x ) ≥ f ′ ( x ) 2 . {\displaystyle f''(x)f(x)\geq f'(x)^{2}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Logarithmically convex function

Start with the simplest possible case. Write down what Logarithmically convex function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Logarithmically convex function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Logarithmically convex function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Logarithmically convex function

In research: Logarithmically convex function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Logarithmically convex function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Logarithmically convex function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Real analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Logarithmically convex function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Logarithmically convex function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Logarithmically convex function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Logarithmically convex function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Logarithmically convex function in simple terms?

In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function. Definition Let X be a convex subset of a real vector space, and let f : X → R be a function taking non-negative val…

Why does Logarithmically convex function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Logarithmically convex function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Logarithmically convex function.