On Inequalities of Trigonometrically ρ- Convex Functions
So, the function is non-decreasing in Case 2. Let and. Differentiating both sides of 5 with respect to one has.
Thus, is non-decreasing function in. Therefore, from Property 2. Theorem 3. To prove the sufficiency, let be an arbitrary point in and has a supporting function at this point. For convenience, we shall write the supporting function in the follwoing form:. For all choose any such that and with and let. Applying 6 twice at and at yields. Multiplying the first inequality by the second by and adding them, we obtain.
Golʹdberg, A. A. (Anatoliĭ Asirovich)
Remark 3. Using 7 at the function becomes. But from 8 ,we observe for all. Hence, the minimum value of the function. If and. The proof mainly depends on Lemma 2. So, we show that the function satisfies all conditions in this lemma. It is obvious that,.
Khlebopros, Rėm Grigorʹevich
First, we study the behavior of the function inside the interval. It is clear from 12 that s is an absolutely continuous function, has a derivative of third order.
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But from the periodicity of and 13 , we get. Using the following substitution. It follows that, can be written as. From the definition of in 14 and the periodicity of we observe that and.
Thus, from 15 and 19 , one has , and. L Dorofeev Book 2 editions published in in Russian and held by 6 WorldCat member libraries worldwide. Populjacionnaja dinamika lesnyh nasekomyh Book 1 edition published in in Russian and held by 2 WorldCat member libraries worldwide. E Ershov Book 1 edition published in in Russian and held by 1 WorldCat member library worldwide.
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