Results & Lemmas (6)
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THEOREM 2.1.
THEOREM 2.1. Let yiy i = 1, 2,, n, be linearly independent solutions of (2.1), let (2.3) /i = -^-, ••-,/.-! = -^=^- Vn Vn
THEOREM 2.1. Let yiy i = 1, 2, , n, be linearly independent solutions of (2.1), let (2.3) /i = -^-, ••-,/.-! = -^=^- Vn Vn
THEOREM 3.1.
THEOREM 3.1. Let y19 y29, yn be linearly independent solutions of (2.1), and let f = yjyn, i — 1, 2,, n — 1. Then the differential equation…
THEOREM 3.1. Let y19 y29 , yn be linearly independent solutions of (2.1), and let f = yjyn, i — 1, 2, , n — 1. Then the differential equation (2.1) is disconjugate in D if and only if every nontrivial linear combination of f, f2, — ,fn^ is (n — l)-valent in D, i.e., it does not take on any one value more than n — 1 times in D.
LEMMA 3.1.
LEMMA 3.1. Let y be analytic in a region R. If ^(α^) = 0, α^ e R, i = 1, 2,, n, then (3.1) yM(z) = Σ (y * ^Pί kj i k = 0, 1,, n — 1, where…
LEMMA 3.1. Let y be analytic in a region R. If ^(α^) = 0, α^ e R, i = 1, 2, , n, then (3.1) yM(z) = Σ (y * ^Pί kj i k = 0, 1, , n — 1, where = Π
THEOREM 3.2.
THEOREM 3.2. Let p0,, pn_y be analytic in the unit disk D = z: <l. If (3.2) Σ (1 + y ^^Y^Ί then the differential equation (3.3) y w +…
THEOREM 3.2. Let p0, , pn_y be analytic in the unit disk D = {z:\z\<l}. If (3.2) Σ (1 + y ^^Y^Ί then the differential equation (3.3) y w + Pn^^)!/^-^ + + po(z)y - 0 is disconjugate in D.
Theorem 3.2 · coeff
Theorem 3.2 generalizes a result recently obtained by Hadass [2, Th. 2], There are known to the author a few other disconjugacy criteria…
Theorem 3.2 generalizes a result recently obtained by Hadass [2, Th. 2], There are known to the author a few other disconjugacy criteria for higher-order equations with analytic coefficients [4, 6]. We are now ready to state the disconjugacy condition (Theorem 3.2) as a multivalence criterion. From Theorems 2.1 and 3.1 we see that every nontrivial linear combination of f,f2, •• ,/w_i is (n — 1)- valent if the equation y {n) + Pn~2{z)y {n~ 2) +
THEOREM 3.3.
THEOREM 3.3. Let f, /2, ••,/%_1 be analytic in the unit disk D = z: < 1. Define pQ, px, -, pn^2 as in (2.4). If det (ff)ZjU does not vanish…
THEOREM 3.3. Let f, /2, •• ,/%_1 be analytic in the unit disk D = {z: \z\ < 1}. Define pQ, px, - , pn^2 as in (2.4). If det (ff)ZjU does not vanish in D, and if (n — k)\ + - V ( l - l*|)(l + \z\Y~'\po{z) then every nontrivial linear combination of f, f2, , fn_γ is (n — 1)-