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Since L is not origin-symmetric and (tauin -1,1)), tauin -1,1rollary 3.2 we know (widethuse{Omega}_{-p}(widehat{nabya }^{tau}_{p}L)>widetilde{Omega}_{-p}(L)).
Since L is not origin-symmetric, Theorem 2.B has (widetilde{G}_{-p}(widehat{nabla}_{p}^{tau}L)> widetilde {G}_{-p}(L)) for (tauin -1, 1)).
Since L is not origin-symmetric and (tau in -1, 1)), in -1llows from Theorem 2.B that (widetilde {G}_{-p}(widehat{nabla}_{p}^{tau}L)> widetilde{G}_{-p}(L)).
(1.12) If K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality if and only if (tau= 0) and equality in the right inequality if and only if (tau= pm1).
Thus, from Theorem 3.A it follows that if K is not origin-symmetric and p is not an odd integer, then equality holds in the right-hand side inequality of (1.13) if and only if (tau=pm1).
(1.10) If K is not origin-symmetric and p is not an odd integer, then there is equality in the left inequality if and only if (tau=0) and equality in the right inequality if and only if (tau=pm1).
(1.8) If K is not origin-symmetric, equality holds in the left inequality if and only if (tau=0) and equality holds in the right inequality if and only if (tau=pm1).
Since L is not origin-symmetric, for (tauin -1, 1)), by Corollary 3.2 we know widetauin -1ega}_{-p}bigl(widehat{nabla}^{tau}_{p}L bigr)>widetilde{Omega}_{-p}(L).
Let (Kinmathcal{S}^{n}_{o}) and (0< p<1), if K is not origin-symmetric, then there exists (Linmathcal{S}^{n}_{os}) such that I_{p}Ksubset I_{p}L, but V K)> V L).
This, together with Theorem 3.3, yields that if K is not origin-symmetric and p is not an odd integer, then equality holds in the left-hand side inequalities of (1.11) and (1.12) if and only if (tau=0).
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