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On l’Hospitaltype rules for monotonicity
 J. Inequal. Pure Appl. Math
"... ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio r: = f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ: = f ′ /g ′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for li ..."
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ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio r: = f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ: = f ′ /g ′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for limits, since in general we do not require that both f and g, or either of them, tend to 0 or ∞ at an endpoint or any other point of (a, b). Here new insight into the nature of the rules for monotonicity is provided by a key lemma, which implies that, if ρ is monotonic, then ˜ρ: = r ′ · g 2 /g ′  is so; hence, r ′ changes sign at most once. Based on the key lemma, a number of new rules are given. One of them is as follows: Suppose that f(a+) = g(a+) = 0; suppose also that ρ ↗ ↘ on (a, b) – that is, for some c ∈ (a, b), ρ ↗ (ρ is increasing) on (a, c) and ρ ↘ on (c, b). Then r ↗ or ↗ ↘ on (a, b). Various applications and illustrations are given.
“NONSTRICT” L’HOSPITALTYPE RULES FOR MONOTONICITY: INTERVALS OF CONSTANCY
, 2008
"... Let f and g be differentiable functions defined on the interval (a,b), where − ∞ � a < b � ∞, and let r: = f g and ρ:= It is assumed throughout that g and g ′ do not take on the zero value anywhere on (a,b). The function ρ may be referred to as a derivative ratio for the “original ” ratio r. In ..."
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Let f and g be differentiable functions defined on the interval (a,b), where − ∞ � a < b � ∞, and let r: = f g and ρ:= It is assumed throughout that g and g ′ do not take on the zero value anywhere on (a,b). The function ρ may be referred to as a derivative ratio for the “original ” ratio r. In [11], general “rules ” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. In particular, according to [11, Proposition 1.9 and Remark 1.14], one has the dependence of the monotonicity pattern of r ( on (a,b)) on that of ρ (and also on the sign of gg ′ ) as given by Table 1. The vertical double line in the table separates the conditions (on the left) from the corresponding conclusions (on the right). ρ gg ′ r> 0
L’HospitalType Rules for Monotonicity and Limits: Discrete Case
, 2005
"... Let − ∞ ≤ a < b ≤ ∞, let f and g be continuously differentiable functions defined on the interval (a,b), and let r = f/g and ρ = f ′ /g ′. In [13], general “rules” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. For example, according to Proposition 1.9 i ..."
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Let − ∞ ≤ a < b ≤ ∞, let f and g be continuously differentiable functions defined on the interval (a,b), and let r = f/g and ρ = f ′ /g ′. In [13], general “rules” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. For example, according to Proposition 1.9 in [13], one has the following:
L’HOSPITALTYPE RULES FOR MONOTONICITY, AND THE LAMBERT AND SACCHERI QUADRILATERALS IN HYPERBOLIC GEOMETRY
 JOURNAL OF INEQUALITIES IN PURE AND APPLIED MATHEMATICS
, 2005
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Coefficients estimates in subclasses of the Carathéodory class related to conical domains
"... Abstract. We study some properties of subclasses of of the Carathéodory class of functions, related to conic sections, and denoted by P(pk). Coefficients bounds, estimates of some functionals are given. 1. ..."
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Abstract. We study some properties of subclasses of of the Carathéodory class of functions, related to conic sections, and denoted by P(pk). Coefficients bounds, estimates of some functionals are given. 1.
ON CONFORMAL REPRESENTATIONS OF THE INTERIOR OF AN ELLIPSE
"... Abstract. We consider the conformal mappings f and g of the unit disk onto the inside of an ellipse with foci at 1 so that f(0) = 0; f 0(0)> 0; g(0) = −1 and g0(0)> 0: The main purpose of this article is to show positivity of the Taylor coecients of f and g about the origin. To this end, we ..."
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Cited by 3 (2 self)
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Abstract. We consider the conformal mappings f and g of the unit disk onto the inside of an ellipse with foci at 1 so that f(0) = 0; f 0(0)> 0; g(0) = −1 and g0(0)> 0: The main purpose of this article is to show positivity of the Taylor coecients of f and g about the origin. To this end, we use a special relation between f and g and the fact that f satises a secondorder linear ODE. Some applications are given to the class of kuniformly convex functions. 1.
Topics in special functions III
 In Analytic Number Theory, Approximation Theory, and Special Functions
, 2014
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ON CONFORMAL REPRESENTATIONS OF THE INTERIOR OF AN ELLIPSE
"... Abstract. We consider the conformal mappings f and g of the unit disk onto the inside of an ellipse with foci at 1 so that f(0) = 0, f 0(0)> 0, g(0) = 1 and g0(0)> 0. The main purpose of this article is to show positivity of the Taylor coecients of f and g about the origin. To this end, we u ..."
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Abstract. We consider the conformal mappings f and g of the unit disk onto the inside of an ellipse with foci at 1 so that f(0) = 0, f 0(0)> 0, g(0) = 1 and g0(0)> 0. The main purpose of this article is to show positivity of the Taylor coecients of f and g about the origin. To this end, we use a special relation between f and g and the fact that f satises a secondorder linear ODE. Some applications are given to the class of kuniformly convex functions. 1.