let g be a graph. in this paper, we study the eccentric connectivity index, the new version ofthe second zagreb index and the forth geometric–arithmetic index.. the basic properties ofthese novel graph descriptors and some inequalities for them are established.

Some new reverses of the Cauchy-Bunyakovsky-Schwarz integral inequality for vector-valued functions in Hilbert spaces that complement the recent results obtained in [1] are given. Applications for the Heisenberg inequality are provided.

Some new reverses of the Cauchy-Bunyakovsky-Schwarz inequality for n−tuples of real and complex numbers related to Cassels and Shisha-Mond results are given.

A reverse of the Cauchy-Bunyakovsky-Schwarz integral inequality for complex-valued functions and applications for the finite Fourier transform are given.

The Cauchy-Schwarz, Buzano and Kre\u{\i}n inequalities are three about inner product. main goal of this article is to present refinements Cauchy-Schwarz inequalities, a new proof refined version Kre\u{\i}n-type inequality. Applications that include Buzano-type for certain operators, operator norm numerical radius Hilbert space operators will be presented.

In the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. In particular, it isshown that the Cauchy-Schwarz inequality holds. Moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy Hilbert space has a complementary subspace.Finally, the notions of fuzzy bo...