[Paper Review] L'Hospital-type rules for monotonicity and limits: Discrete case

Iosif Pinelis|arXiv (Cornell University)|Nov 21, 2006

Mathematical Inequalities and Applications12 references3 citations

TL;DR

This paper establishes discrete analogues of L'Hospital-type rules for monotonicity and limits, deriving conditions under which the ratio of discrete functions $ r = f/g $ exhibits specific monotonicity patterns—such as decreasing then increasing—based on the monotonicity of the ratio of differences $ \rho = \Delta f / \Delta g $ and the sign of $ g \Delta g $. The key contribution is a complete classification of monotonicity behavior of $ r $ in terms of $ \rho $ and $ g\Delta g $, with explicit characterization of turning points via supremum and infimum constructions.

ABSTRACT

Assuming that a "derivative" ratio rho:=f'/g' of the ratio r:=f/g of differentiable functions f and g is monotonic (that is, rho is increasing or decreasing), it was shown in previous papers that then r can switch at most once, from decrease to increase or vice versa. In the present paper, "discrete" versions of such l'Hospital-type rules for monotonicity (as well as "discrete" versions of l'Hospital's rules for limits) are obtained, for functions f and g defined on an interval of integers.

Motivation & Objective

• To extend continuous L'Hospital-type rules for monotonicity to discrete sequences.
• To identify conditions under which the ratio $ r = f/g $ of discrete functions exhibits a 'decreasing-then-increasing' or 'increasing-then-decreasing' pattern.
• To provide a systematic classification of monotonicity behavior of $ r $ based on the monotonicity of $ \rho = \Delta f / \Delta g $ and the sign of $ g\Delta g $.
• To offer a practical framework for determining the monotonicity pattern of $ r $ without requiring limits at endpoints.

Proposed method

• Define discrete functions $ f_n $, $ g_n $ on integer intervals $ \overline{a,b} $, with $ g_n, \Delta g_n \neq 0 $ and constant sign.
• Define $ r_n = f_n / g_n $ and $ \rho_n = \Delta f_n / \Delta g_n $, where $ \Delta f_n = f_n - f_{n-1} $.
• Establish a key identity: $ g_n g_{n-1} \Delta r_n = (\rho_n - r_n) g_n \Delta g_n = (\rho_n - r_{n-1}) g_{n-1} \Delta g_n $, linking differences of $ r $ to $ \rho $ and $ r $.
• Use extremal constructions (e.g., $ k = \sup\{n : \Delta r_n \leq 0\} $) to prove existence of turning points in monotonicity patterns.
• Apply symmetry and reflection arguments (e.g., $ n \mapsto -n $, $ f \mapsto -f $) to reduce cases and strengthen conclusions.
• Use the identity and sign analysis to derive contradictions when monotonicity assumptions are violated, proving the classification in Table 1.

Experimental results

Research questions

• RQ1 Under what conditions does the ratio $ r = f/g $ of discrete sequences exhibit a single turning point, transitioning from decreasing to increasing or vice versa?
• RQ2 How does the monotonicity of $ \rho = \Delta f / \Delta g $ and the sign of $ g\Delta g $ jointly determine the monotonicity pattern of $ r $?
• RQ3 Can the monotonicity pattern of $ r $ be fully classified without requiring $ f $ and $ g $ to tend to 0 or $ \infty $ at endpoints?
• RQ4 What is the precise role of the sign of $ g\Delta g $ in determining whether $ r $ is $ \searrow\nearrow $ or $ \nearrow\searrow $?
• RQ5 How can one distinguish between the cases where the extremum of $ r $ occurs at an endpoint or in the interior of the interval?

Key findings

• If $ \rho $ is nondecreasing and $ g\Delta g > 0 $, then $ r $ is nonincreasing on $ \overline{a,k} $ and nondecreasing on $ \overline{k,b} $ for some $ k \in \overline{a,b} \cup \{a,b\} $, i.e., $ r \searrow\nearrow $.
• If $ \rho $ is nonincreasing and $ g\Delta g > 0 $, then $ r $ is nondecreasing on $ \overline{a,k} $ and nonincreasing on $ \overline{k,b} $, i.e., $ r \nearrow\searrow $.
• When $ \rho $ is nondecreasing and $ g\Delta g < 0 $, the monotonicity pattern of $ r $ is $ \nearrow\searrow $, and when $ \rho $ is nonincreasing and $ g\Delta g < 0 $, it is $ \searrow\nearrow $.
• The existence of such a $ k $ is guaranteed by the supremum construction $ k = \sup\{n \in \overline{a+1,b} : \Delta r_n \leq 0\} $, with $ \Delta r_n > 0 $ on $ \overline{k+1,b} $.
• The proof uses a contradiction argument based on the key identity: $ g_n g_{n-1} \Delta r_n = (\rho_n - r_n) g_n \Delta g_n $, showing that violating monotonicity leads to contradiction with $ \rho $'s monotonicity.
• For $ \alpha > 0 $, the sequence $ r^{(\alpha)} = f^{(\alpha)}/g $ with $ f_n^{(\alpha)} = \alpha + \sum_{j=0}^n p_j $ and $ g_n = \sum_{j=0}^n q_j $, where $ \rho = p/q $ is increasing, satisfies $ r^{(\alpha)} \searrow\nearrow $, with the minimum at $ k_\alpha $, and $ \alpha_k = (\rho_k - r_k^{(0)}) g_k $ is the threshold where $ \Delta r_k^{(\alpha_k)} = 0 $.